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Research in Industrial Projects for Students

Institute for Pure & Applied Mathematics

University of California, Los Angeles

Sponsor

The Aerospace Corporation

Final Report

Calculating Call Blocking, Preemption

Probabilities and Bandwidth Utilizationfor Satellite Communication Systems

Student Members

Leah Rosenbaum, [email protected] (Project Manager)

Mohit Agrawal

Leah Birch

Yacoub Kureh

Academic Mentor

Nam Lee, [email protected]

Sponsoring Mentors

James Hant, [email protected]

Eric Campbell, [email protected]

Krista O’Neill, [email protected]

Brian Wood, [email protected]

James Gidney, [email protected]

Consultants

Amber Puha, IPAM Associate Director

August 26, 2011

This project was jointly supported by The Aerospace Corportation and NSF Grant 0931852

Abstract

Many satellite communication (SATCOM) systems currently use a static resource alloca-tion scheme, which is known to provide excellent service to high-priority jobs. However, thisstatic allocation method is inefficient, as it makes valuable network resources inaccessibleto low-priority jobs even when resources are available. We developed stochastic modelsthat approximate the SATCOM systems and studied their statistical properties. We beganby reviewing the standard M/M/m model from classic queueing theory and used Markovchains to extend the classic models to include multi-class jobs, more suitably approximatingSATCOM systems. Then, by way of simulation, we isolated and characterized the system’skey performance trends in terms of its statistical characteristics. Properties under consid-eration include the probability of call blocking, preemption, satisfaction, and the expectedserver utilization, all as a function of priority and bandwidth.

Keywords: queueing theory, satellite communications, communication systems, Erlangloss, loss system, stochastic processing network, multi-class queueing models

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Acknowledgments

We are very grateful for the help of James Hant, Eric Campbell, Krista O’Niell, BrianWood, and James Gidney of The Aerospace Corporation and our academic mentor NamLee for their invaluable support, feedback and direction on this project.

We would also like to extend our gratitude to Mike Raugh and the rest of the IPAM andRIPS staff for their extensive support throughout the RIPS program.

We also appreciate the contributions of Amber Puha to our project.

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Contents

Abstract 3

Acknowledgments 5

1 Introduction 131.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Models 172.1 Performance Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Single-Type Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Competing Priority Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Competing Bandwidth Classes . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Results 253.1 Single-Type Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Competing Priority Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 Competing Bandwidth Classes . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Comparison of Static and Dynamic Resource Allocation Methods 414.1 Competing Priority Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Competing Bandwidth Classes . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Conclusion and Future Work 455.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

REFERENCES

Selected Bibliography Including Cited Works 49

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List of Figures

1.1 Simple SATCOM System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 The System as a Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Priority System as Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Bandwidth System as Markov Chain . . . . . . . . . . . . . . . . . . . . . . 23

3.1 Single-Type: M/M/1 System Occupancy . . . . . . . . . . . . . . . . . . . . 263.2 Single-Type: M/M/4 System Occupancy . . . . . . . . . . . . . . . . . . . . 263.3 Single-Type: M/M/m Blocking Probability . . . . . . . . . . . . . . . . . . 273.4 Single-Type: Multiplot: M/M/m Expected System Occupancy . . . . . . . 273.5 Single-Type: Multiplot: M/M/m Blocking Probability . . . . . . . . . . . . 283.6 Priority: Simulation vs. Theory . . . . . . . . . . . . . . . . . . . . . . . . . 293.7 Priority: High Priority Blocking . . . . . . . . . . . . . . . . . . . . . . . . . 303.8 Priority: High Priority Server Utilization . . . . . . . . . . . . . . . . . . . 303.9 Priority: Low Priority Blocking and Preemption Probabilities . . . . . . . . 313.10 Priority: Blocking and Preemption Probabilities vs. Ratio λH to λL . . . . 323.11 Priority: Preemption vs. ρtotal . . . . . . . . . . . . . . . . . . . . . . . . . 323.12 Priority: Preemption vs. Ratio λH to λL . . . . . . . . . . . . . . . . . . . . 333.13 Priority: Total System Server Utilization and Statistic vs. ρtotal . . . . . . . 333.14 Bandwidth: Server Utilization: λsmall = 30 and λbig = 1 . . . . . . . . . . . 343.15 Bandwidth: Server Utilization: λsmall = 20 and λbig = 2 . . . . . . . . . . . 353.16 Bandwidth: Server Utilization: λsmall = 10 and λbig = 3 . . . . . . . . . . . 353.17 Bandwidth: Total Server Utilization vs Traffic Intensity . . . . . . . . . . . 363.18 Bandwidth: Total Server Utilization vs Ratio . . . . . . . . . . . . . . . . . 363.19 Bandwidth: Blocking Probability vs Ratio for Big Bandwidth Jobs . . . . . 373.20 Bandwidth: Blocking Probability vs Ratio for Small Bandwidth Jobs . . . . 383.21 Bandwidth: Trade Plot for Big Bandwidth Jobs . . . . . . . . . . . . . . . . 393.22 Bandwidth:Trade Plot for Small Bandwidth Jobs . . . . . . . . . . . . . . . 39

4.1 Priority Comparison: Satisfaction . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Priority Comparison: Server Utilization . . . . . . . . . . . . . . . . . . . . 424.3 Priority Comparison: Satisfaction vs. Utilization . . . . . . . . . . . . . . . 434.4 Bandwidth Comparison: Satisfaction . . . . . . . . . . . . . . . . . . . . . . 444.5 Bandwidth Comparison: Server Utilization . . . . . . . . . . . . . . . . . . 44

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List of Tables

1.1 Static and Dynamic Resource Allocation Methods . . . . . . . . . . . . . . 14

3.1 Bandwidth: The Robustness of Total Traffic Intensity ρtotal . . . . . . . . . 35

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Chapter 1

Introduction

1.1 Objectives

In this report, we consider variants of classic queueing models to better understand theperformance of satellite communication systems which must deal with job requests that havedifferent priority statuses. Our study is motivated by the current focus of The AerospaceCorporation, a federally funded, non-profit research and development center. One of TheAerospace Corporation’s primary functions is the development, launch, and maintenance ofsatellites. As part of their work with satellites, The Aerospace Corporation helps coordinatesatellite communication systems.

Figure 1.1: Simple SATCOM system. Satellites have a limited capacity to satisfycommunication circuits.

Satellite communication (SATCOM) systems have a limited capacity to satisfy commu-nication circuits. For example, the satellite in Figure 1.1 only has 100 Mbps of capacity.A key problem in the management of SATCOM systems is the allocation of the limitedcommunications resources among competing users, who have pre-assigned priority statusesand have differing bandwidth needs. In Figure 1.1, the red circuit represents a high-prioritycommunication which only requires 256 kbps of bandwidth, while the green circuit repre-sents a low-priority communication with requires 1.544 Mbps of bandwidth. Designing anefficient policy is not only a challenging and time-consuming task but also a costly venturein terms of research and implementation.

Currently a static allocation method is used to allocate resources among users. That

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(a) Potential outcome of the current static resource allocation method, which leaves servers under-utilized and the lower priority users unserved. Table based on table from June 28th presentationby The Aerospace Corporation at IPAM.

(b) Potential outcome of dynamic resource allocation policy, which would better utilize the serversand allow lower priority users access to the system. This method also introduces blocking andpreemption. Table based on table from June 28th presentation by The Aerospace Corporation atIPAM.

Table 1.1: Static and Dynamic Resource Allocation Methods

is, users are pre-assigned access to the SATCOM system based on their priority statuses.However, static allocation is often inefficient. As illustrated in Table 1.1a, high-priorityusers are often given their own reserved channel, regardless of their usage pattern or needs.Also, users can reserve their channels according to the maximum foreseeable bandwidthrequirements irrespective of the overall use. This results in two detrimental effects: manylower priority users are blocked from system access, and the system remains underutilized.

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In this table, no low-priority users were able to access the system. Further, the system isunderutilized–for example, in the first time period, only 75 Mbps of this system is utilized,leaving 25 Mbps free. However, the two low-priority users who were unable to access thesystem in this time period had only requested a total of 16 Mbps. Thus, a more efficientallocation of system resources would have been to give access to these low-priority users bytaking away access from higher-priority users that are not actually using the system in thattime period.

Addressing the inefficiencies of the static allocation method could create a usage patternsimilar to that in Table 1.1b. In this allocation scheme, users are not given preassignedaccess to the system. Instead, resources are allocated dynamically as jobs enter and leavethe system. Such a dynamic allocation method results in higher utilization and reduces thenumber of users who are blocked from service.

Our team’s task is to conduct an exploratory analysis to understand the potential benefitand cost of using a dynamic allocation policy instead of a static allocation policy. Upon thecompletion of the project, The Aerospace Corporation would have access to the computerprograms written in MATLAB that were used to complete the numerical experiments forour exploratory analysis.

Finally, our results do not constitute a recommendation for The Aerospace Corporationto adopt a specific allocation method. Instead, our work represents a first step that will beextended by The Aerospace Corporation to be more applicable to their SATCOM systems.

1.2 Approach

While other possibilities exist, our work focuses on two key aspects of SATCOM systems:jobs with different priority levels and jobs with different resource requirements.

In order to take these two aspects into account, we reduce the SATCOM system into acircuit-switching network1 with m available circuits but with no queue buffer. This allowsus to treat the SATCOM system as equivalent to the classic M/M/m/02 queueing model(alternatively, we term this the “single-type” model, because all elements in the system havethe same priority status and bandwidth requirements). In the M/M/m model, arriving jobsenter the system if there is at least one free server; else those jobs are blocked. When a jobin the server is processed completely, it leaves the system, and the resources it was usingbecome available for other jobs.

The classic M/M/m model is then extended to handle jobs of different priorities, and tohandle jobs of different bandwidths. Lastly, we compared two dynamic allocation schemes–one based on the differing priorities model, and the other based on the differing bandwidthsmodel–to the current static allocation method.

1Many SATCOM systems are circuit-switching networks rather than packet-switching networks.In a circuit-switching network, two users must have a direct connection to one another for communi-cation to occur–no other users may be using their circuit. On the other hand, in a packet-switchingnetwork, packets from different communicants can commingle as they are transmitted through thesystem. Thus, the space of allocation strategies for circuit-switching networks differs from strategiesfor packet-switching networks.

2We will often omit the last 0, and thus talk about the “M/M/m model”. This expression isan example of Kendall’s notation, which is a compact description of queueing models of the formα/β/m/N [2]. α and β refer to the arrival and service processes, respectively; m describes thenumber of servers; and N is the maximum queue length. Often, the arrival and departure processesare Markovian, and are thus denoted with M.

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We highlight the cost and benefits of the various allocation schemes in terms of keyperformance measures3 such as call blocking, preemption probabilities, and bandwidth uti-lization4. In practice, the call blocking and preemption probabilities were combined into aperformance measure called user satisfaction. An important plot in our analysis, then, isthe trade plot between user satisfaction and bandwidth utilization (as the arrival intensityand bandwidth m of the system change). The trade plots, along with other plots of per-formance measures, form the crux of our analysis; from these plots, we are able to describeoverall trends and behavior of the single-type model, the competing priority classes model,and the competing bandwidth classes model.

1.3 Overview

The remainder of our report is as follows: In Chapter 2, we briefly review elementaryconcepts of stochastic processing networks. From these concepts, we develop performancemeasure for the three types of system: the single type traffic, the competing priority classesmodel, and the competing bandwidth classes model. In Chapter 3, we present our results,identifying trends in the performance measure developed in Chapter 2 over ranges of inputsto the system. In Chapter 4, we compare the results of our dynamic allocation system tothe same measures of a static allocation scheme, first for competing priority classes, thenfor competing bandwidth classes. Finally, in Chapter 5, we present our conclusions andsuggestions for future work. We summarize overall trends from each of the three systemtypes and address areas for further investigation.

3The performance measures were calculated using both theoretical and simulation approaches.In the theoretical approach, a Markov chain was used to model the three different systems. In thesimulation approach, each system was modeled using discrete event simulation.

4In the single-type model, these performance measures depend on the intensity of jobs enteringthe system, and the amount of bandwidth m available in the system. In the multi-class models, theperformance measures also depend on the ratio of arrival rates of competing classes.

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Chapter 2

Models

This chapter is organized as follows: we first introduce performance measures that we use tocharacterize and compare the systems operating under different policies. This is done underan M/M/m setup, but we will pay particulate attention to the performance measures’ rel-evance to developing efficient policies for a SATCOM model. We then explain performancemeasures for systems that include priority service and various bandwidth requirements–altering the measures from their M/M/m definitions or eliminating them where appropri-ate.

2.1 Performance Measures

To evaluate system performance under different policies, we consider the following per-formance measures: call blocking and preemption probabilities, and server or bandwidthutilization. A call is blocked when there are not enough servers available in the systemto handle the job. Preemption occurs only in the priority model, and is described morefully there (see Section 3.3). Server utilization describes the average system utilization–howmuch bandwidth is being occupied on average.

These measures are tracked as a function of varying traffic intensity, ρ, which is theratio of overall arrival and departure rates from the system, and the overall bandwidth ofthe system m. The exact formula will change for different types of systems.

2.2 Single-Type Model

For one job type, we can consider a stochastic processing network with m servers and queuesof length 0. As such, there can be at most m jobs in the system at any time. If a job seeksto enter the system but no free servers are available, the job is blocked. Because blockedjobs are lost forever, this system is known as the Erlang loss system.

The system passes from state x1 to state x2 whenever a new job arrives or a job leavesthe system. Once the system is in a given state, the probability of it entering another stateis fixed and independent of the system’s past states (see Figure 2.1). In other words, wemodel a stochastic processing network as a finite state, continuous time Markov process[15].

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Figure 2.1: The System as a Markov Chain

Infinitesimal Transition Rate Matrix

Our assumptions about the interarrival and service time distributions lead to a compactdescription of the underlying Markov process. For example, when the underlying operatingpolicy is FIFO, an M/M/m model can be completely described, at least in terms of itsprobabilistic properties, by the following matrix:

Λ =

−λ λ . . . 0 0 . . . 0 0µ −(λ+ µ) . . . 0 0 . . . 0 00 2µ . . . 0 0 . . . 0 0...0 0 . . . −(λ+ (m− 1)µ) λ . . . 0 00 0 . . . mµ −(λ+mµ) . . . 0 00 0 . . . 0 mµ . . . 0 00 0 . . . 0 0 . . . −(λ+mµ) λ0 0 . . . 0 0 . . . mµ mµ

where “(i, j)” for i = j, and the diagonal elements specify the rate of leaving state i in aninfinitesimally short period of time. Such a matrix can be defined uniquely. Hence, Λ issaid to be “the” infinitesimal transition rate matrix.

Stationary Distribution

In this report, we only work with systems that are irreducible 1 and stationary 2. In particu-lar, the limiting distribution (i.e. the long run time-average) and the stationary distributionare unique, and one and the same. Moreover, computation of the stationary distributioncan be reduced to a system of linear equations. That is, the stationary distribution π is theunique solution to the following equations:

πTΛ = 0, (2.1)

πT1 = 1, (2.2)

1Irreducible: Any state i can be reached from a given state j in a finite number of steps.2A stochastic process {X(t), t ≥ 0} is said to be a stationary process if for all n, s, t, . . . , tn the

random vectors X(t1), . . . , X(tn) and X(t1 + s), . . . , X(tn + s) have the same joint distribution [15].

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where xT denotes the transpose of x, and Λ is the infinitesimal rate matrix for the under-lying Markov process. In other words, from matrix Λ, we can find the system’s stationarydistribution with regard to occupancy state. When the model is both irreducible and re-current,3 each πi is the same as the proportion of time that the system is in state i oncethe system is at steady state. Clearly, all the πi sum to 1.

Arrival/Service Distributions

Interarrival times are distributed exponentially with mean 1/λ, and the amount of time ajob seizes server resources - its service time - is also distributed exponentially with mean1/µ. The minimum of the service times is exponential with parameter equal to pµ, wherep is the number of jobs currently being served. Jobs leave the system, meaning that thesystem passes from state x to state x− 1, with rate xµ.

For these systems, traffic intensity is defined as

ρ =λ

mµ.

System Occupancy

System occupancy is the stationary distribution for a specific state. For 0 ≤ i ≤ m,

P(the system is in state i) = πi =

(λ/µ)i

(i!)m∑j=0

(λ/µ)j

j!

. (2.3)

Blocking Probability

Blocking occurs only when the system is fully occupied, that is, in the m state. Blockingprobability can be read off as a special case of system occupancy.

P(being blocked) = πm =

(λ/µ)m

(m!)m∑j=0

(λ/µ)j

j!

. (2.4)

2.3 Competing Priority Classes

Next, we integrate priority classes into our M/M/m model, allowing two competing pri-orities, high and low. Even when all server bandwidth resources are being used, a highpriority job can enter the server so long as there is at least one low priority job present;that lower priority jobs is preempted, or removed, from the server, and the high priorityjob gains access to their resources. The preempted low priority job is considered lost to thesystem; they do not get the opportunity to resume service or re-enter the server.

3For any state i, let fi denote the probability that, starting in state i, the process will ever reenterstate i. The system is recurrent if fi = 1 for all i [15].

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For simplicity, we only consider systems with server capacity m = 100 bandwidth units.Each job class has its own arrival rate λH for high priorities and λL for low priorities, whichwhen added together give the total arrival rate, λtotal. The classes have identical servicerate µ. The total traffic intensity is defined as

ρtotal = ρH + ρL =λH + λL

mµ. (2.5)

To build intuition, consider a system with m = 1. There are three possible system occu-pancy states: 0 occupancy, 1 low priority job, or 1 high priority job. The state transitionscan be modeled as a Markov chain (Figure 2.2).

Figure 2.2: Priority System as Markov Chain

The resulting infinitesimal transition rate matrix is:

Λ =

−(λL + λH) λL λH

µ −(µ+ λH) λH

µ 0 −µ

.

As in the non-preemptive queueing background, πTΛ must equal 0. Since the πi also sumto 1, we can solve the system of equations and find the following state probabilities:

π0 =µ

λH + λL + µ, πlow =

µλL

(λH + µ)(λH + λL + µ), πhigh =

λH

λH + µ. (2.6)

For higher m-values, we use software (such as MATLAB or Mathematica) to calculateinfinitesimal transition rate matrix and the resulting stationary state probabilities.

Note that classifying occupancy states by the number of occupants is no longer sufficient.In the classic M/M/m model, there was only one state that contained 4 occupants. In thepriority model, there are five different states in which the system has 4 occupants: 4 lowpriority, 3 low priority and 1 high priority, 2 low priority and 2 high priority, 1 low priorityand 3 high priority, 4 high priority. To clarify, we will now use the notation π(H,L) whereH,L give the number of high and low priority jobs in the system.

In addition to the performance measures used for M/M/m, we also measure preemptionprobability. Preemption occurs when a low-priority job is removed from the system beforecompleting its service. We define user satisfaction to be the percent of total jobs thatcompleted their service without being blocked or preempted.

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These performance measures are tracked in terms of the total traffic intensity and ratio ofarrivals between the two priority classes. The bandwidth m is kept constant, and departurerates, µ, are held at 1. Occasionally, measures are also compared to M/M/m systems withidentical arrival and departure rates to judge the effect of priority.

System Occupancy and Utilization

The system occupancy distribution is still the proportion of time the system spends ineach occupancy state, a measure we can consider by total occupants or by priority class.From occupancy, we can also determine the system utilization or mean server utilization,by calculating expected value of system occupancy, either by each class or for both classescombined. For instance, the high priority server utilization is

∑iπ(i,L),

where 0 ≤ i ≤ m and L is any number of low priority jobs, such that i+ L ≤ m.

Blocking Probability by Priority

Ideally, high priority jobs always receive service. They are only blocked at very high trafficrates - that is, when the system is quickly filled by other high priority jobs. Theoretically,high priority behavior mimics that of a non-preemptive M/M/m system with the parame-ters of the high priority class. These results should match π(m,0), the probability that thepreemptive system is full of high priority jobs.

Low priority jobs lack preemptive privileges and are blocked whenever the server is fullyoccupied, regardless of the priority status of those jobs in the system. Thus, the probabilityof low priority blocking is

∑(H,L)∈S

π(H,L),

where S = {(H,L) : H + L = m}.

Preemption Probability

A new performance measure for the priority system is the probability of a low priority jobbeing preempted from the server. Suppose all server resources are in use. If a job usingresources is low priority, an incoming high priority job can seize those resources and remove(i.e. preempt) that low priority job from the system. Low priority jobs are preempted if ahigh priority job arrives to the system when the system is in a full state - that is, all serverresources are in use - and at least one of the jobs in the system is low priority. We nowsketch the heuristic derivation of the preemption probability. The probability of being in afull state with at least one low priority job is∑

(H,L)∈S

π(H,L),

where

S = {(H,L) : H + L = m and L ≥ 1}. (2.7)

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The probability that a high priority job enters the system is λHλH+λL

. Then the product

∑(H,L)∈S

π(H,L)λH

λH + λL

yields the probability of preemption out of the total number of arrivals. We want theprobability that a low priority job is preempted, so we need to multiply the expected numberof jobs arriving to the system and divide again by the expected number of low priority jobsarrivals to the system. Thus, the probability that a low priority job is preempted from thesystem is:

∑(H,L)∈S

π(H,L)λH

λH + λL

λH + λL

λL=

∑(H,L)∈S

π(H,L)λH

λL.

Satisfaction Probability

In a preemptive priority system, we are interested in the percentage of arrivals that completetheir requested service. The probability of being satisfied is the probability that a job wasnot blocked or preempted.

Satisfaction = 1− (Blocking probability + Preemption probability).

Like other performance measures, we can separate satisfaction probability into three dif-ferent categories: high priority, low priority, and total system. Since low priority jobs canbe preempted, their probability of being satisfied is highly affected by the presence of highpriority jobs. Certain SATCOM systems currently have very low satisfaction levels for lowpriority jobs, so this measure is of particular interest.

2.4 Competing Bandwidth Classes

Our next system allows jobs to have different bandwidth requirements requirementsB1, . . . , BK . Should a job be assigned a bandwidth that exceeds the bandwidth resourcescurrently available in its server, that job queues until those resources become available.4.The degenerate case is of a system in which all jobs request a constant x bandwidth andtotal bandwidth is mx–this is precisely identical to the M/M/m model.

Throughout this report, we discuss the simplest nontrivial bandwidth system: a systemwith two job classes with different bandwidth requests. We can intuitively think of this asa communication system having a certain bandwidth capacity and handling two types ofjobs, for example telephone calls which require some small amount of bandwidth and videocalls which require some larger amount of bandwidth.

To develop intuition for the bandwidth model, consider a system with a server capacityof 4 bandwidth units and 2 job classes: those jobs requiring 1 bandwidth and those jobsrequiring 2 bandwidth5. If the jobs arrive with rates λ1 and λ2, respectively, and both

4In our models, assigned bandwidth requirements never exceed the maximum server capacities.5Any scaling of these three numbers results in an equivalent system, i.e m = 4x, B1 = x, and

B2 = 2x where x ∈ R+ are equivalent.

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Figure 2.3: Bandwidth System as Markov Chain

have service rate µ, the system can be described by the Markov chain6 in Figure 2.3. Notethis chain is two dimensional, which corresponds to the number of job classes. The orderedpair (n1, n2) indicates the state of system having n1 jobs of first class (those requesting 1bandwidth) and n2 jobs of the second class (those requesting 2 bandwidth). We impose theordering7 on the ordered pairs from lowest n1 to highest and lowest n2 to highest in orderto make the following infinitesimal transition rate matrix:

(0,0) (0,1) (0,2) (0,3) (0,4) (1,0) (1,1) (1,2) (2,0)(0,0) -(λ1 + λ2) λ1 0 0 0 λ2 0 0 0(0,1) µ -(µ+ λ1 + λ2) λ1 0 0 0 λ2 0 0(0,2) 0 2µ -(2µ+ λ1 + λ2) λ1 0 0 0 λ2 0(0,3) 0 0 3µ -(3µ+ λ1) λ1 0 0 0 0(0,4) 0 0 0 4µ −4µ 0 0 0 0(1,0) µ 0 0 0 0 -(µ+ λ1) λ1 0 0(1,1) 0 µ 0 0 0 µ -(2µ+ λ1) λ1 0(1,2) 0 0 µ 0 0 0 2µ -(3µ) 0(2,0) 0 0 0 0 0 2µ 0 0 −2µ

As before, we use this transition matrix to find the stationary probability distribution vectorπ. However, now the ordering of the vector is not intuitive as we had to impose an orderingon the states, that is elementπi corresponds to some state (n1, n2) which depends on ourordering above. For example, π6 denotes π(1,1). We begin our πi indexing at zero as we didin classic M/M/m, so that π0 always corresponds to the empty system state. Again, notethat there is not some natural ordering but we must impose one to compute the stationaryprobability distribution vector.

The performance measures discussed are server utilization, blocking probability, andsatisfaction. These performance measures are collected both in aggregate and for each jobclass. Performance measures will be parameterized by the total traffic intensity defined later(3.1) and the ratio of arrival rates. The ratio of arrival rates will be weighted by bandwidth,

that isB1λ1

B2λ2. Total bandwidth in the system, m, is kept constant, as in the service rate

parameter µ for all job classes.

6Although the state space can be infinite if m = ∞ or if there are infinitely many different Bi,the number of possible states is finite when working with a finite set of bandwidth types, {Bi}, anda finite bandwidth system m < ∞. When the state space is finite, the Markov chain is ergodic, andthus a stationary distribution exists.

7This ordering is arbitrary and only imposed for computational purposes.

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Server Utilization

Server utilization is similar to system or server occupancy discussed in the classic model.It differs in that it no longer counts the number of jobs in the system but rather countsthe bandwidth used in the system. Again, we calcualte this measure by simply multiplyingthe number of jobs of class i, ni, by the amount of bandwidth requested by class i, Bi.Rigorously, the average server utilization of the total system is∑

(n1B1 + n2B2 + n3B3 + . . .+ nKBK)π(n1,n2,n3,...,nK).

To calculate the average server utilization for a particular job class, i, we use the formula∑(niBi)π(n1,n2,n3,...,nK).

Blocking Probability and Satisfaction

Blocking probability remains identical to that in the previous models. If a job requesting xbandwidth attempts to join the system, and there is less than x bandwidth available, thatjob will be blocked. Thus, during heavy server utilization, bigger jobs face more difficultyentering the system than smaller jobs because it is less likely that sufficient resources areavailable.

A job from a class requesting bandwidth x is blocked when the system is in state(n1, n2, n3, . . . , nK) where m − (n1B1 + n2B2 + n3B3 + . . . + nK) < x. Therefore theblocking probability for class i is ∑

(n1,n2,n3,...,nK)∈S

π(n1,n2,n3,...,nK) (2.8)

where S = {n1, n2, n3, . . . , nK) : m− (n1B1 + n2B2 + n3B3 + . . .+ nK) < Bi}.To calculate the total blocking probability of the system, we weight each class by its

bandwidth amount and arrival rate. Let βi denote the blocking probability for a class i asdefined in equation (2.8). The total blocking probability therefore is∑

(βiBiλi)∑(Biλi)

.

As there is no priority in this model, satisfaction is simply the complement of blockingprobability. We introduce this performance measure to add consistency rather than provideany new insight.

24

Chapter 3

Results

The results of our work are organized by the system analyzed: M/M/m, preemptive pri-ority with a single bandwidth requirement, and multiple bandwidth requirements withoutpriority. All these systems have Markovian arrival and departure processes and do not havea queue. The M/M/m system has a server capacity of variable m units. As discussed inChapter 2, system occupancy and blocking probability are the measures for these systems.In a preemptive priority, single bandwidth requirement system, arriving jobs have identicalbandwidth requirements and fall into one of two competing priority classes. Performancemeasures for this system are blocking and preemption probabilities, server utilization, andsatisfaction. In a system with multiple bandwidth requirements and without priority, ar-riving jobs have identical priority but fall into one of two competing bandwidth classes. Forthese systems, we consider only server utilization and blocking probability.

3.1 Single-Type Model

Before we add features such as bandwidth and priority, we must test our simulation methodagainst the M/M/m theoretical model to ensure the accuracy of our underlying model. Wefirst examine the simplest case of m = 1, but we also vary m to test the robustness ofour simulation. For the performance measures, we compared the simulated results to thetheoretical predictions developed in the methods section using the χ2 goodness-of-fit test 1,giving the p-values listed by the figures.

3.1.1 System Occupancy

The first of our results is illustrated by a histogram for system occupancy for m = 1,meaning the system could hold 1 job when occupied and 0 jobs at all other times. Plottedin green in Figure 3.1 is the theoretical value for system occupancy, calculated from Formula2.3.

As Figure 3.1 indicates, the simulation results matched the theoretical results very welland were well within the confidence interval. The results also make intuitive sense as ρchanges; when ρ > 1, the arrival rate is higher than the service rate, the requests thatreceive service take longer to get that service, and the distribution favors the system being

1The χ2 goodness-of-fit test passes for p-values greater than 0.05. In this test, the null hypothesisis that the distributions match, and the alternative hypothesis is that the distributions do not match.A p-value greater than 0.05 indicates that the null hypothesis should be accepted.

25

0 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

SimulationTheoryConfidence Interval

(a) ρ = 2, p-value = 0.97464

0 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7


(b) ρ = 1, p-value = 0.96182

0 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7


(c) ρ = 0.5, p-value = 0.93892

Figure 3.1: M/M/1 System Occupancy by Proportion of Total Time for VaryingTraffic Intensity

occupied. When ρ = 1, arrival rate and service rate parameters are equal, so the systemis equally likely to be in either of the two states. Finally, when ρ < 1, requests arrive ata much slower frequency than they are served, there is a considerable amount of idle timebetween services, and the system is more likely to be in the unoccupied 0 state.

Next, we consider how the system operates for m values greater than one. We lookspecifically at m = 4 in the following graphs, but other m values yield similar trends.As in the M/M/1 case, the strong relation between simulation and theoretical predictionsremains, with all p-values at or very near 1. Again, it is more probable for the system tobe in higher occupancy states when ρ > 1 and lower states when ρ < 1. Here, note that atρ = 1 in Figure 3.2b, the states are not all equally likely. The 4 servers make the rate ofleaving state 4 slower than the rate of leaving state 3 slower than the rate of leaving state2 etc. as these rates are exponential with means 1/4µ, 1/3µ, 1/2µ respectively. Thus thehigher occupancy states within the server are somewhat more likely at ρ = 1. Again noticethat this trend follows the theoretical value from Formula 2.3.

0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SimulationConfidence IntervalTheory

(a) ρ = 2, p-value =0.961

0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


(b) ρ = 1, p-value = 0.999

0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


(c) ρ = 0.5, p-value = 0.996

Figure 3.2: M/M/4 System Occupancy for Varying Traffic Intensity

3.1.2 Blocking

The Monte Carlo estimate of blocking probability (Figure 3.3) adheres closely to equation2.4. Predictably, the probability of being blocked increases as ρ increases since, as Figures3.1 and 3.2 emphasize, the system favors the occupied state.

26

As expected, adding more servers significantly decreases the blocking probability. Notethat in the M/M/8 system in Figure 3.3b, a ρ = 1 gives about 25% chance of blocking whilethe same ρ in an M/M/1 system (Figure 3.3a) gives about 50% blocking. As ρ increases,the m = 1 and m = 8 systems appear more similar in terms of blocking probability, butthere is still a 15% difference at a traffic intensity of 2.

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Traffic Intensity, ρ

Pro

babi

lity

of B

lock

ing

Blocking Probability vs. Traffic Intensitym = 1, N = 0

TheorySimulation

(a) m = 1

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Traffic Intensity, ρP

roba

bilit

y of

Blo

ckin

g

Blocking Probability vs. Traffic Intensitym = 8, N = 0

TheorySimulation

(b) m = 8

Figure 3.3: M/M/m Blocking Probability by Traffic Intensity

3.1.3 Overall Trends

We took care to note overall trends in theM/M/mmodel so we can evaluate how competingpriority and bandwidth classes affect performance. Our simulation matches theoreticalvalues, so for the sake of time, the following plots are based on theory.

0 0.5 1 1.5 20

1

2

3

4

5

6

7

8

9

10


Exp

ecte

d O

ccup

ancy

Sta

te

Expected System Occupancy vs. Traffic Intensitym = 1, 2, ... 10, N = 0

m = 1m = 2m = 3m = 4m = 5m = 6m = 7m = 8m = 9m = 10

Figure 3.4: Multiplot: M/M/m ExpectedSystem Occupancy by Traffic Intensity forVarying m

We consider values of m = 1 to 10 tolearn how the system reacts to varying mvalues as the traffic intensity increases. InFigure 3.4, we examine how system occu-pancy increases with m. System occupancyincreases steadily with ρ as the server fillsup. However, near ρ = 1, occupancies be-gin to increase at a decreasing rate as thesystems reach maximum capacity. With ρabove 1, the systems separate fairly clearlyby m value as they are likely to be com-pletely occupied.

In the blocking probability multiplot(Figure 3.5), the pattern from Figure 3.3is extended and filled in. The higher m-value systems clearly provide lower blockingprobability, though the performance differ-ence decreases as the number m of serversincreases. For example, increasing the number of servers from 1 to 3 shows significant im-provement, but the improvement associated with increasing the number of servers from 6 to

27

10 is less drastic, particularly as traffic intensity increases. A comparison of blocking prob-ability and system occupancy reveals that a higher occupancy corresponds with a higherblocking probability; with more jobs filling up the system, there is a greater chance of thesystem being full when an arrival occurs.

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Pro

babi

lity

of B

lock

ing

Blocking Probability vs. Traffic Intensitym = 1,... 10 & 10000, N =0

m = 1m = 2m = 3m = 4m = 5m = 6m = 7m = 8m = 9m = 10m=10000

Figure 3.5: Multiplot: M/M/m BlockingProbability by Traffic Intensity for Varyingm

28

3.2 Competing Priority Classes

Figure 3.6: Simulation vs. Theory withp-value= 0.9996

To understand the behavior of the com-peting priority classes model, we firstconducted simulation trials and comparedthose results to those for the M/M/msingle-type model. We discovered that thesystem occupancy for high priority jobsmatches the theory from M/M/m exactly,but low priority jobs do not. Therefore,for analyzing our performance measures, wegraphed M/M/m theory to (1) illustratethe similarities present in the entire systemand (2) the differences preemption causes inthe performance of low priority jobs.

We later developed a theory for preemp-tive priority policy using Markov chains.Again, we used a χ2 test to numerically as-sess the goodness-of-fit between theory and simulation. Figure 3.6 illustrates the closematch between the simulated and theoretical results with a p-value of 0.9996, showing thevalidity of our theory.

3.2.1 High Priority Jobs

For the high priority job class, we consider blocking probability and server utilization2.Total system performance measures are considered later. Performance measures for highpriority jobs are measured against the following parameters: total traffic intensity (ρtotal),high priority traffic intensity (ρH), and the ratio of high priority to low priority arrivals (λH

: λL).

High Priority Blocking

High priority arrivals are blocked from entering the system only when the server is fullyoccupied by other high priority jobs. Therefore, as high priority traffic intensity increases,high priority blocking should also increase, as the following graphs illustrate.

Figure 3.7a plots high priority blocking against high priority traffic intensity for variousratios of λH to λL. While Figure 3.7a gives the expected result of increased blocking withincreased traffic intensity, it also indicates that the ratio of high priority to low priorityjobs does not affect high priority blocking. So long as the high priority jobs have the sameρH , they experience the same blocking probability. Figure 3.7a also plots the M/M/100with the arrival rate of λH . Note that high priority blocking follows the dashed M/M/100theory line exactly. Based on the plotted lines, we conclude that high priority jobs enterthe system without regard for low priority jobs.

In Figure 3.7b, we consider high priority blocking probability as a function of the ratioof λH to λL; each line represents a different total traffic intensity. With more arrivals overall(the red and light blue lines), blocking is more likely. As the ratio of λH to λL increasesalong the horizontal axis, high priority jobs begin to block other high priority arrivals. In

2Recall that server utilization for the high priority class refers to the amount of server resourcesseized by high priority jobs.

29

(a) Blocking vs. Traffic Intensity (b) Blocking vs. Ratio of Arrivals

Figure 3.7: High Priority Blocking

Figure 3.7b, this happens when λH to λL is close to 1, and the blocking probability continuesto increase with the arrival rates ratio.

High Priority Server Utilization

Figure 3.8: High Priority Server Utiliza-tion

We also consider the average proportionof server resources seized by high prior-ity jobs, a measure which gives a sense ofthe resources available to low priority jobs.Figure 3.8 plots the mean server utiliza-tion for high priority jobs against ρH . In-tuitively, higher traffic intensity increasesthe mean high priority server utilization.Again, this graph emphasizes that high pri-ority requests follow the same trends as theM/M/100 model, as high priority jobs areunaffected by the low priority job class.

Considering Figures 3.7a and 3.8 in tan-dem reveals the trade-off between server uti-

lization and blocking. Both server utilization and blocking probability increase as ρtotalincreases; the more of the server that is utilized, the more likely it is that jobs are blocked.

3.2.2 Low Priority Jobs

The low priority class has similar metrics to the high priority class, but in addition toblocking, low priority jobs may also be preempted. This addition means that unlike highpriority jobs, low priority jobs should not follow the M/M/m theory exactly, except incases where they greatly outnumber the high priority jobs (meaing preemption is not verylikely). We speculate that in some cases, low priority jobs receive limited service, so wealso calculated the percentage of low priority jobs that complete service–the low prioritysatisfaction probability.

30

Low Priority Blocking and Preemption

Whether a low priority job is blocked from the system or preempted, the result is the same:the low priority requested service is not completed. Therefore, it is helpful to consider thesetwo measures side by side.

(a) Low Priority Blocking (b) Low Priority Preemption

Figure 3.9: Low Priority Blocking and Preemption Probabilities

Figure 3.9a shows that low priority jobs are increasingly blocked as the total trafficintensity increases. Since low priority blocking relies on the total traffic intensity3, the sameproportion of low priority jobs are all blocked for different ratios of high priority arrivals tolow priority arrivals. We also have M/M/100 plotted on the graph for an extreme λL. Lowpriority jobs only mimic M/M/100 when almost no high priority jobs arrive, such as thearrival ratio of 1 λH to 1000 λL.

In Figure 3.9b we can see how the preemption probability differs from blocking prob-abilities that we have seen before; unlike the blocking probabilities that increase with thetraffic intensity, the preemption probability increases, but only to a point before droppingoff. For low traffic intensities, the server is never filled, so preemption is low. Eventually,the system becomes overwhelmed by arrivals, at which point high priority arrivals preemptthose low priority jobs in the server. Preemption reaches a peak and then declines because,after a certain point, there are no low priority jobs in the system to be preempted; theyare simply blocked. M/M/100 is not plotted on the preemption graph because there is nopreemption in a classic stochastic networking system. Low priority preemption is higherwhen there are more high priority arrivals than low with the highest preemption probabilitycorresponding to the greatest number of high priority arrivals. When there are more lowpriority requests than high priorities, preemption stays rather low.

We also plotted preemption and preemption + blocking probability against the ratioof λH to λL for different total traffic intensities. The resulting graphs reinforce the trendsobserved in previous preemption plots.

As the ratio of high priority requests to low priority requests increases, the preemptionprobability within a total traffic intensity levels off (Figure 3.10a). For ρtotal = 1/2 , nopreemption occurs because the system is never filled. Systems with ρtotal = 1 and 3/2 followthe same trend of increasing blocking probability. However, ρtotal = 2 has a distinct changein the maximum preemption probability, as the maximum probability is less than that of

3This is because low priority traffic can be blocked both by other low priority jobs and highpriority jobs.

31

(a) Low Priority Preemption (b) Low Priority Blocking + Preemption

Figure 3.10: Low Priority Blocking and Preemption vs. Ratio of λH to λL

lower ρtotal values. We would suspect that there would be more preemption because of thegreater ρtotal, but in fact there is a trade off between blocking and preemption as Figure3.10b illustrates. There is more blocking and less preemption at a certain traffic intensity,so overall fewer low priority jobs are served. It is interesting to note that blocking of lowpriority jobs does not depend on λH to λL. For a given traffic intensity, blocking remainsconstant, and the lines in Figure 3.10b only increase at the point where preemption beginsto occur, which we can see in Figure 3.10a.

Low Priority Server Utilization

Figure 3.11: Preemption vs. ρtotal

We analyze server utilization by low priorityjobs to see how high priority jobs affect lowpriority jobs. In Figure 3.11, low priorityjobs seize more server resources when highpriority requests arrive less frequently (thetop lines in the plot). A system with manymore low priority arrivals than high priority(the top black line) also corresponds to thedashed M/M/100 theory line. The lack ofhigh priority jobs means that low priorityjobs enter and use the server more often.

Most systems climb to a peak server uti-lization before decreasing; this trend is mostobvious in λH/λL = 1 (the middle blue

line). When ρtotal > 1, the mean server utilization of low priority jobs decreases. Theincreased ρtotal means increased ρH , which in turn means increased preemption and/orblocking. The decline in low priority server utilization is more dramatic when the ratio ofλH to λL is one. Furthermore, mean server utilization is lowest when high priority arrivalrates are higher than low priority.

32

Low Priority Satisfaction

Figure 3.12: Preemption vs. Ratio λH toλL

Low priority requests are less likely to re-ceive service due to preemption. There-fore, their probability of being satisfied isdependent on the ratio of λH to λL. Fig-ure 3.12 plots satisfaction against the ratioof arrival rates for various ρtotal values. Forρtotal = 1/2, all low priority jobs are sat-isfied because the system never completelyfills with jobs. However, as ρtotal increases,the satisfaction probability decreases. Thenas high priority arrivals increase relative tolow, the probability of a low priority job be-ing served decreases further. For systemswith ρtotal = 3/2, 2 and a ratio of λH toλL greater than one, low priority jobs areblocked or preempted more often than theyare served.

3.2.3 Total System Statistics for the Priority Model

(a) Total Server Utilization (b) Probability of Being Satisfied

Figure 3.13: Total System Server Utilization and Statistic vs. ρtotal

In Figures 3.13a and 3.13b, the varying ratio of λH to λL does not change server uti-lization or satisfaction probability; the system as a whole matches the M/M/100 model forρtotal. High and low priority jobs act differently depending on the arrival rate ratio, but thetotal system statistics are those of an M/M/100 system.

Figure 3.13a illustrates that as ρtotal increases, so does the total server utilization. Moretraffic means that more of the server is necessary to serve the increasing number of incomingjobs. When ρtotal > 1, we can see that the system approaches complete server utilization.Total satisfaction is plotted in Figure 3.13b, which emphasizes that a higher percentage ofjob are satisfied when ρtotal < 1. The probability of being satisfied decreases as the amountof traffic increases. The higher number of arrivals means that more jobs are likely to beunsatisfied.

33

3.3 Competing Bandwidth Classes

In the competing bandwidth classes model, the classic concept of traffic intensity, ρ, must be

redefined. We compute an analogous total traffic intensity, calculated as ρtotal =∑ λiBi

mµi.

As a simplification, we assume that all service rates are equal, µi = µ. We ignore those jobclasses in which Bi ≥ m, as it will always be blocked.

ρtotal =∑Bi<m

λiBi

mµi. (3.1)

We will use this definition of total traffic intensity for all systems with competing band-width classes. As a matter of completeness, it is important to point out that other moresophisticated definitions do exist. Kelly discusses other “effective bandwidths” [8] [9] thatserve a similar purpose4

3.3.1 Preliminary Results

In the following graphs, we ran the simulation with a bandwidth set B = {Bsmall, Bbig},where Bsmall = 1, Bbig = 10, and m = 100. Again, we return to our intuitive example ofthe telephone call and video call job classes. The system has 100 bandwidth units and eachtelephone call (the small bandwidth class) uses 1 unit while video calls (the big bandwidthclass) uses 10 units. As before, the green line represents the theoretical result derived usinga two dimensional Markov chain.

Figure 3.14: Bandwidth: Server Utiliza-tion: λsmall = 30 and λbig = 1

In Figures 3.14, 3.15, and 3.16, ρtotalis a constant 40

100 = .4 where µ = 1 andBsmallλsmall +Bbigλbig = 1λsmall + 10λbig =40. The ratio

λsmall

10λbig(3.2)

is varied between graphs. Again, the ratio isweighted by bandwidth because we measurethe utilization on a per bandwidth basis. In-tuitively, when ratio (3.2) is greater thanone, server utilization comes predominantlyfrom the small jobs, as they are arriving (ina per bandwidth sense) more quickly. Theserver utilization histogram is “smoother”

and unimodal, as bandwidth states jump by increments of one, rather than ten (Figure3.14). When ratio (3.2) equals 1, the server utilization is split evenly among the big andsmall jobs. Utilization jumps of size ten are more frequent than in the previous graph, sothe modality increases (Figure 3.15). The reason for this increases modality is a bit subtle,but its reason is intuitive: binning.

When working with the classical model we binned by increments of one, because serverswere seized in increments of one. If we were to have binned in increments of 0.5, our graphswould clearly have been multi-modal but for a trivial reason. Here, similarly, the multi-modality comes from binning; a lot of jobs are seizing ten servers at a time. In binning by

4Kelly also uses effective bandwidths to describe admission control policies for systems withcompeting bandwidth classes.

34

ones (in order to show the entire state space), we inadvertently introduce multi-modality.Thus, we should expect that when ratio (3.2) is less than 1, the utilization histogram be-comes distinctly multi-modal because the big bandwidth jobs frequenty increase utilisationby increments of ten (Figure 3.16).



Now consider extreme values of this ra-tio. When ratio (3.2) nears ∞, there areinfinitely more small jobs or essentially nobig jobs arriving, so the the system has onlyone job class that uses 1

100 of the system’sbandwidth. Thus the ratio produces anM/M/100 system at ∞. On the other ex-treme, a ratio of 0 produces the M/M/10case for the same reason.

It is important that we recognize thatmulti-modality is not an inherent feature ofthe simulation or the theory. It is merely anartifact of how we view the data and resultsfrom binning.

Next we investigate the proposed defi-nition of traffic intensity. For small ρ, weexpect that mean server utilization wouldequal ρ. When ρtotal = 0.4, mean serverutilization for four systems is listed in Ta-ble 3.1; we note that mean utilization in-deed centers around 40, or 40% utilization.Utilization volatility increases with λbig asthese jobs cause bigger jumps in bandwidthutilization. This volatility affects meanserver utilization in that, as λbig increases,the difference between the expected utiliza-tion (calculated to be 40 based on ρtotal)and the observed utilization grows.

It is also important to note that our sim-ulation and theory match very well with ap-value over 0.99 using a χ2 test in Figures

3.14, 3.15, and 3.16. Our simulation provides an intuitive way to see how the system runsover time and reaches its steady state.

λbig λsmall Mean Server Utilization Mean Utiliza-tion - (ρtotal×m)

% Error

1 30 40.0119 0.0119 0.032 20 40.0440 0.0440 0.113 10 39.8370 0.1630 .414 0 39.7659 0.2341 0.59

Table 3.1: Bandwidth: The Robustnessof Total Traffic Intensity ρtotal

35

Theory, on the other hand, provides a mathematically rigorous solution to the simulation’sproblem of run-time efficiency. Although the simulation is important in that allows moreflexibility, all forthcoming results are derived strictly from theoretical calculations.

3.3.2 Server Utilization

In the above section we looked at a few examples of bandwidth models. We will continue tofocus on systems with 100 bandwidth and job classes B1=1 and B2 = 10 with service ratesµ = 1 and no queues. However, now instead of focusing on certain cases, we investigate

overall trends of the system as a function of two parameters, ρtotal and the ratioB1λ1

B2λ2=

λ1

10λ2. We may sometimes refer to the classes as small and big instead of 1 and 2, respectively

We investigate trends in server utilization. In aggregate, this measure indicates whetheror not the system is being utilized efficiently. That is, can it handle more traffic, or is itover-utilized? We plot utilization in two ways: against ρtotal with different lines for different

ratiosλ1

10λ2and against this ratio with different ρtotal values. Although not materially

different, these two graphs provide different insights into these parameters. For these graphs,

0.1 ≤ ρtotal ≤ 2 and 10−4 ≤ λ1

10λ2≤ 102.

Figure 3.17: Total Server Utilization vsTraffic Intensity

Figure 3.18: Total Server Utilization vsRatio

In Figure 3.17, the first thing to noteis that all the lines are coincident for smalltraffic intensities. In underutilized systems,there is no effective difference between theratios. There are two reasons for this: thesystem has significant bandwidth remain-ing, so there is little blocking. Also, ρtotal isidentical across the different ratios, mean-ing that the amount of bandwidth beingrequested per unit time is also equivalentacross the ratios. The lines begin to divergewith higher ρtotal. Looking just at the dot-ted lines, we notice the familiar trend: theclassic M/M/100 system outperforms theM/M/10 system in that the M/M/100 sys-tem is more utilized for the same traffic in-tensity, implying that the system is moreefficient.

Note that certain ratios perform worsethan an M/M/10 system. As expected, theline for ratio 10−4 (the black line) is es-sentially coincident with the M/M/10 linethroughout, however, as the ratio increases(that is more small jobs come into the sys-tem), we see that the performance actuallydeclines before it gets better and eventu-ally surpasses M/M/10 and converges withM/M/100 line as we should expect for largeratio (which means almost all arriving jobs

36

are small bandwidth). This is one of the most important trends found in the bandwidthmodel.

Before exploring it further, consider the plot versus ratio in which this trend is far morepronounced (Figure 3.18). In the case where ρtotal = 0.5 (the bottom, black line), there islittle change as the ratio varies. However, ρtotal = 2 (the top, yellow line), server utilizationfalls to a minimum at 10−2 and then rises to the theoretical maximum of the M/M/100as the ratio shifts from favoring large bandwidth jobs on the left to small bandwidth onthe right. There is a slight oscillation in the graph. We saw before that the M/M/10 caseperforms worse than M/M/100, but the dip in the middle is not immediately intuitive. Wewill reason it out slowly.

It is important to remember is that this trend applies only in heavy traffic systems. Ina low ratio case (approximated by M/M/10), the big bandwidth jobs compete only withthemselves. Utilization changes primarily by increments of 10 with the two most commonstates being those with 90 or 100 bandwidth units used. As small jobs begin arriving morequickly, the system spends time in states with 91 − 99 bandwidth units being utilized.When the system is in one of these states, jobs of size ten are blocked until all of the smallbandwidth jobs leave the system or until one of the big bandwidth jobs leaves.

It takes longer for all the small jobs to leave the system than for just one of the bigbandwidth jobs to leave. This is an intuitive matter, but it can also be worked out rigorouslyusing probability theory. A big bandwidth job would likely enter the system as result of bigbandwidth job leaving. However, there are already fewer big jobs in the system by virtueof there being small bandwidth jobs in the system, so the expected waiting time for one bigbandwidth job leaving the system is significant. This trend becomes more pronounced asthe ratio increases until server utilization begins to increase at the ratio of 10−2. The reasonthis turn around occurs here becomes clearer when we will look at trade plots. The takeaway message from these plots is that, in terms of server utilization, certain mixed ratiosystems perform worse than pure systems where big bandwidth jobs compete only againstthemselves.

3.3.3 Blocking Probability

The following figures plot blocking probability against the arrival rates ratio.

Figure 3.19: Blocking Probability vs Ratiofor Big Bandwidth Jobs

Figure 3.19 tracks blocking for the largebandwidth jobs with different lines for dif-ferent values of ρtotal. Notice that for theheavy traffic intensity lines (ρtotal ≥ 1), in-creasing the arrival rate ratio (bringing inmore small jobs relative to large) increasesbig bandwidth blocking dramatically. Asmentioned above, small jobs are very effi-cient at occupying just enough of the systemto block large bandwidth jobs. For under-utilized systems (ρtotal = .5), big jobs actu-ally have decreased blocking probability asthe ratio increases likely because the smallbandwidth jobs use the system more effi-ciently. With fewer large bandwidth jobsentering utilization volatility is decreased.The small bandwidth jobs keep the system’s

37

utilization average closer to 50%, thus decreasing the chances that the system enters stateswith 91 to 100 bandwidth units in use. Consequently, blocking probability for big bandwidthjobs is lower.

The red line for ρtotal = .75 illustrates both of these effects. On the left side of the graph,the line matches M/M/10, as we would expect for a small ratio. Moving right, blockingprobability increases slightly due to volatility from the big bandwidth class. This effectdiminishes as the ratio grows and more small bandwidth jobs use the system, decreasingblocking probability for the big bandwidth jobs.

Figure 3.20 focuses on the blocking probability for the small bandwidth class. Blockingprobability for this job class should never exceed the blocking probability for the big band-width class for any fixed ρtotal and ratio. This restriction follows from the calculation ofblocking probability (2.8). First consider the extreme ratios. As expected, blocking followsthat of M/M/100 on the right side of the graph where the ratio is high.

Figure 3.20: Blocking Probability vs Ratiofor Small Bandwidth Jobs

On the other extreme, the blockingprobability of small jobs is equivalent tothat for M/M/10. At this extreme, the sys-tem rarely contains any small bandwidthjobs. When blocking occurs, it is becausethe system is 100% utilized, meaning thatany kind of job is blocked. Most notablein this the behavior at intermediate ratios.Surprisingly, even in high traffic systems,small bandwidth jobs have effectively noblocking probability at certain arrival ra-tios.

As the ratio increases, small bandwidthjobs gain access to the system. The systemspends more time in states with 91 - 95%utilization. After this point, big bandwidthjobs will rarely dominate the system again, and small bandwidth jobs effectively get thoselast ten units of bandwidth for themselves. With a ratio of 10−2 and ρtotal = 2 (the yellowline), λ1 is around 2. Consider those last nine or ten bandwidth units as a separate systemreserved for small bandwidth jobs. Within this single class system, ρ ≈ 0.2, and we expectlow blocking probability. The minor oscillations in this plot will be addressed later. Theimportant trend to notice is that for mixed ratios, small jobs have a significantly lowerblocking probability than in systems where small jobs would only compete with themselves.

3.3.4 Trade Plots

Similar to previous trade plots, Figures 3.21 presents the trade off between server utilizationand blocking probability for the large bandwidth class. The different lines represent differentratios of weighted arrival rates. This plot “begins” in the bottom left corner where ρtotal ≪1, and both mean server utilization and blocking probability are near zero. From here,each line goes in a different direction until ultimately reaching the top left corner whereρtotal ≫ 1, blocking probability is unity, and mean server utilization becomes nearly zeroagain. Consider the blue line with ratio 0.1. Both jobs arriving at equal rates, but clearlybig bandwidth jobs arrive more quickly on a per bandwidth basis.

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Figure 3.21: Trade Plotfor Big Bandwidth Jobs

Just by increasing ρtotal, the blue linegoes from the bottom left toward the upperright corner and then back toward the upperleft corner. The early behavior is to be ex-pected. With increasing ρtotal, we expect tosee increasing mean server utilization sim-ply because more big bandwidth jobs enterthe system and seize resources. We also ex-pect blocking probability to increase, as thesystem spends more time in full states. In-terestingly, after a certain point of increas-ing traffic intensity, the mean server utiliza-tion by big jobs plummets back to zero. Bigbandwidth jobs are blocked by small band-width jobs.

This turning point occurs at different places for different ratios but is persistent regard-less of ratio (except for the dotted line that corresponds to M/M/10). This plot is slightlydeceptive as it does not show ρtotal values along the lines. In order to get the black lineto bend back around, we had to increase ρtotal to the pragmatically implausible value of30, 000! For normal traffic intensities, we get the expected behavior of increased utilizationand blocking probability with increased ρtotal. However, it is theoretically possible to setthe ρtotal high enough to cause this reversing behavior, pushing all big bandwidth jobs fromthe system.

Supplementary Results

Figure 3.22: Trade Plotfor Small Bandwidth Jobs

In Figure 3.22, we show the trade plot forsmall bandwidth jobs, and in it we seethe most pronounced oscillatory behavioryet. The lines corresponding the ratios{10−2, 10−3, 10−4} have very apparent os-cillations. As we saw in the trade plot forbig bandwidth jobs, these plots correspondto unrealistically large values for ρtotal, butthey serve the purpose of exaggerating theoscillatory behavior, and thus provide use-ful insight. Let’s focus on the red line cor-responding to the ratio 10−3. This line goesstraight up as the blocking probability in-creases, yet no small bandwidth jobs are inthe system.

Since ρtotal is about 100 at this point,the arrival rate for the small bandwidth job is slightly less than 1 while the arrival rate forbig bandwidth jobs is approximately 99.9. Clearly the system will to be completely utilized.It will be incredibly rare for a small job to enter the system. If one does happen to enter, itsservice rate is exponentially distributed with a mean 1, which is smaller than mean arrivalrate for small jobs. Thus in the steady state, small jobs will not remain in the system.

Once the arrival rate for small bandwidth jobs increases past 1, small bandwidth jobscontrol a larger block of bandwidth. This explains the first hump in our plots, that is,

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as the small bandwidth jobs begin to seize part of the system, their blocking probabilitydrops dramatically, and server utilization increases. As the small job arrival rate increasespast ten, this reserved block of bandwidth no longer suffices, and the small jobs beginto block themselves. As ρtotal increases past 20, another oscillation occurs as the smallbandwidth jobs seize another block of bandwidth. This oscillatory behavior continues butdecays quickly as small bandwidth jobs begin to control the entire system.

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Chapter 4

Comparison of Static and DynamicResource Allocation Methods

In this chapter, we compare two dynamic allocation methods–one based on the differingpriorities model, the other on the differing bandwidths model–with the static allocationmethod currently employed by some SATCOM systems. For these comparisons, we measuremean server utilization and satisfaction. We find that dynamic allocation schemes provideimproved performance overall, but that not all classes of jobs experience improvements.

4.1 Static vs Dynamic Resource Allocation for

Competing Priority Classes

To model static allocation for competing priority classes, we consider a system with aserver capacity of 100 bandwidth units. Portions of this capacity are reserved for eachclass following a high:low ratio. The various lines in the multiplots below indicate systemperformance when the indicated number of bandwidth units is allocated to the indicatedpriority class. Jobs from each class were assumed to arrive with equal frequency λH = λL

and have service rate µ = 1. The total traffic intensity, ρtotal, is still calculated as

ρtotal =λH + λL

mµ.

Figure 4.1 charts the various satisfaction levels for each priority class along differentallocation lines and varying traffic intensities. While all systems provide high prioritysatisfaction above 85%, dynamic allocation provides the highest satisfaction, even slightlyhigher than the 99:1 allocation line. The low priority satisfaction plots (Figure 4.1b) isa bit more complicated. While satisfaction is high initially, it falls quickly with trafficintensity greater than 1. The low level of satisfaction for low-priority users is due to thehigh preemption rate. As indicated in Figure 3.9b, when the classes have equal arrivalrates (the dark blue line), the probability of preemption rises from 10% to above 40% as ρincreases from 1 to 2. While the blue dynamic line is worse than the static allocation lines,it would still outperform a system reserved completely for high priority jobs–that is, whenlow priority jobs receive no service–a distinct possibility under static allocation.

In addition to satisfaction, we compare resource utilization in the static and dynamicresource allocation schemes. As the results in Figure 4.2 indicate, dynamic resource alloca-tion creates a much more efficient system than any of the various static allocation. More

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Figure 4.1: Satisfaction vs Traffic Intensity by Resource Allocation and Priority Class

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Figure 4.2: Mean Server Utilization vs Traffic Intensity by Resource Allocation

specifically, at ρ = 1, the dynamically allocated system sees 95% utilization whereas allthe statically allocated systems see just under 50% utilization. With the same underlyingresources, a dynamic allocation method allows for increased overall utilization, althoughsome of the low priority communication is interrupted (the significant preemption in Figure4.1b).

Dynamic allocation results in a more efficient system. Figure 4.3 shows that it takesa higher traffic intensity to get a static system to a utilization level equal to that of adynamic system since the reserved resources are off limits to the other priority class. Thishigher traffic intensity means that more (usually low priority) jobs are blocked from thesystem, bringing down satisfaction. With a dynamic approach, available resources areshared between all that request them; notice that the green dotted line indicates almostno blocking until the utilization is about 95%. Low priority jobs may be removed from adynamic model in order to better serve high priority jobs, but the overall satisfaction levelis still much improved from any of the static systems.

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Figure 4.3: Total Satisfaction vs Traffic by Total Server Utilization by ResourceAllocation

4.2 Static vs Dynamic Resource Allocation for

Competing Bandwidth Classes

To compare static and dynamic resource allocation methods for competing bandwidthclasses, we divide the 100 bandwidth unit server capacity equally in half. Jobs from eachbandwidth class can only use the resources allocated to their class. We consider two classesof jobs–those that require 1 server unit or 10 server units. For the static case, jobs of size 1effectively enter their own M/M/50 system while jobs of size 10 enter an M/M/5 system1.The arrival rates are such that λ1 = 10λ10, making the bandwidth arrival rates equal. Theservice rates are held constant, and total traffic intensity is still calculated as

ρtotal =λ1 + 10λ10

µ.

As the figures indicate, dynamic allocation yields somewhat mixed results between band-width classes. While both classes enjoy 100% satisfaction under dynamic allocation fortraffic intensities below 0.5, (Figure 4.4), satisfaction for large bandwidth jobs falls sharplyto about 70% at ρ = 1 and is only 12% at ρ = 2. Satisfaction for small bandwidth jobsstays relatively high, reaching a low of about 82% at ρ = 2. In static allocation systems,both classes fall to about 50% satisfaction by ρ = 2. As Figure 4.4c indicates, the overalleffect of dynamic allocation is to slightly increase satisfaction as compared to the staticallocation case. The increase in small bandwidth job satisfaction comes from the ability toblock large bandwidth jobs even when the server is not fully occupied. Once the server has91 out of 100 unit occupied, all large bandwidth jobs are blocked while small bandwidthjobs can still access server resources.

Under the dynamic method, both classes increase their server utilization as ρ increases.Yet large bandwidth utilization peaks at 35-40 of the 100 units and then declines while smallbandwidth utilization grows steadily to about 85 of the 100 units. In the static allocationsystem, each class increases utilization before reaching a plateau at about 45 units. Thegains of the small bandwidth class come at the expense of large bandwidth class utilization.

1As we mentioned in Chapter 3, we can scale systems. A bandwidth model with a single classwhere B = 10 and m = 50 is equivalent to a system with B = 1 and m = 5, which is preciselyM/M/5

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Figure 4.4: Satisfaction vs Traffic Intensity by Bandwidth Class

Figure 4.5c indicates that the small bandwidth class gains are just enough to slightly morethan off-set the large bandwidth class losses, making dynamic resource allocation slightlymore efficient than static allocation for competing bandwidth classes.

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Figure 4.5: Server Utilization vs Traffic Intensity by Bandwidth Class

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Chapter 5

Conclusion and Future Work

5.1 Conclusion

For our exploratory analysis of the potential benefit and cost of a dynamic allocation policyover a static allocation policy, we considered three different stochastic models: the single-type model, the differing priorities model, and the differing bandwidths model. Performancemeasures, including user satisfaction and resource utilization, were analyzed for these threemodels. We then compared two dynamic allocation schemes–one based on the differingpriorities model, and the other based on the differing bandwidths model–to the currentstatic allocation method.

We summarize the results from each of the three traffic models:

Single Traffic Class Jobs that request a smaller fraction of the total bandwidth resourceshave better performance at a given traffic intensity.

Two Competing Priorities (1) High Priority Traffic only competes with itself and itsperformance is determined by the high priority traffic intensity. (2) As the percent-age of high priority jobs increases, more low priority jobs are preempted and theirutilization of bandwidth resources decreases. (3) The combined system performancefor the prioritized system is identical to the non-prioritized system.

Two Competing Bandwidths (1) Total performance is bounded above by the smallbandwidth class. (2) Small bandwidth jobs utilize resources more easily than largebandwidth jobs and can more easily block them from getting service. (3) Systemswith large and small bandwidth jobs arriving at similar rates perform marginallyworse in aggregate than systems in which the large bandwidth jobs compete onlyamonst themselves.

We then compared the two dynamic allocation models with the static allocation model.Dynamic allocation schemes provide better overall performance (resource utilization anduser satisfaction) than comparable static allocation schemes. However, performance variesacross different traffic classes. Dynamic allocation schemes provide the best performancefor high priority and smaller bandwidth jobs. At high traffic intensities, however, dynamicresource allocation allows low priority jobs to be preempted by high priority jobs and highbandwidth jobs to be blocked by small bandwidth jobs. This decreases performance for lowpriority, large bandwidth jobs compared to static allocation approaches.

Our results apply to a simplified SATCOM system with only one satellite and withidealized arrival and service processes. Without specific knowledge of their systems, we

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cannot make definite recommendations for The Aerospace Corporation. However, fromthe work presented here, there appears to be great value in applying a dynamic resourceallocation scheme to certain SATCOM systems, and we recommend that The AerospaceCorporation pursue future research in this area.

5.2 Future Work

The results achieved in this project open several avenues for future research. Some of thisfuture work pertains directly to The Aerospace Corporation and may be undertaken bythem (or by a future RIPS research team!).

Future Work Related to SATCOM Systems

The priority model could be extended to allow for any integer n1 number of priority classes,not just two classes. Similarly, the bandwidth model could be extended to allow for anyinteger n2 number of bandwidth classes1.

Once the priority and bandwidth models have been extended to handle multiple classes,a combined model could be developed that allows jobs to have varying priority statuesor bandwidth requirements. Thus, if the priority model has n1 different classes, and thebandwidth model has n2 different classes, then the combined model would have jobs ofn1 × n2 number of classes.

The model could also be further developed to better describe SATCOM systems.

• Rather than modeling SATCOM systems as consisting of a single satellite, the systemcan be modeled as a group of satellites. This allows for the addition of routing oftraffic through satellite crosslinks and/or terrestrial gateways.

• Alter the distributions for arrival and service processes. Alternative arrival and serviceprocesses may not be Markovian, thus making theoretical analysis more difficult.Simulation results, however, should hold.

• In addition to priority status and bandwidth requirements, a further class character-istic could be the frequency channels required by jobs. Thus, job classes would bedefined by (frequency, priority, bandwidth) triples.

• In order to use a satellite for communications, users must be in the antenna coverageregion of that satellite. Adding antenna coverage and beam pointing to the modelswould make the models more accurate.

• The various dynamic allocation schemes should be tested on duty cycle data fromthe SATCOM systems. Such testing would provide system-specific performance thatgoes beyond general performance results from these stochastic models.

Other Future Work

There are opportunities to explore the behavior of priority models with different preemptionrules.

1In addition, for the differing bandwidths model, the MATLAB simulation code could be extendedto handle non-integer amounts of bandwidth.

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• Our models assume that a preempted job is an unsatisfied job. An alternative modelcould assign gradations of satisfaction to users who achieved partial service.

• Preempted jobs could be put on hold or tagged with slightly higher priority shouldthey reenter the system at a later time. This would allow a comparison between our“preemption-kill” priority model with a “preemption-resume” model. Preemption-resume has been studied in [12] and [19].

• The current priority model can be compared to a nonpreemptive priority model. In anonpreemptive model, low-priority users are not preempted from the server. However,other techniques like admission control (i.e., even if there are empty servers, low-priority users may not be allowed access) may be used in a nonpreemptive model toimprove the performance for high-priority users.

Similarly, the behavior of the competing bandwidths model with different policies couldbe studied.

• In heavy-traffic systems, high-bandwidth jobs are relatively disadvantaged to low-bandwidth jobs. To compensate for this effect, we could give high-bandwidth jobs ahigher priority.

• An alternative to giving high-bandwidth jobs more priority is to adopt an admis-sion control policy for the system. Such a policy, for example, may prevent a low-bandwidth job from entering the system if it would use the 91st server (and wouldthus block any more high-bandwidth jobs). Such admission control policies may besimilar to trunk reservation, as described in [7] and [10].

Another field of research is optimal control. In this case, the goal is to identify anoptimal allocation policy for the SATCOM system. However, one must first choose whichobjective to maximize–user satisfaction, resource utilization, some combination thereof, orother performance measures. One component of optimal control research would be thedevelopment of a flexible, adaptive policy. Such a policy would be able to calibrate thepolicy to account for fluctuations in the arrival intensity of jobs in real-time.

Lastly, the computational calculation of performance measures is very expensive; theprocess of constructing a Markov chain and then solving to find the stationary distributionis an O(N2) operation. In practical terms, systems with more than 2000 states take toolong to solve in MATLAB. To reduce the computational cost, research can be undertaken

• to develop closed-form equations to calculate performance measures, thus avoidingthe intermediate step of calculating the stationary distribution.

• on approximation techniques for the stationary distribution. One approximation, assuggested in [9], is to pre-estimate the stationary distribution, then choose to solvea smaller system in which only the most salient states are kept. Such a truncationtechnique is accurate if the pre-estimate technique works well. Another technique isto use approximations to the matrix exponentiation function, as described in [15].

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Selected Bibliography IncludingCited Works

[1] I. Angus. An Introduction to Erlang B and Erlang C. Telemanagement, 187:6–8, 1983.

An approachable introduction to modeling telecommunication networks as Er-lang loss systems, and to the Erlang B and C equations. Description of software toolsis outdated.

[2] S. Asmussen. Applied Probability and Queue. Wiley Series in Probability andStatistics. John Wiley & Sons Inc, 1987.

Contains much background on classical queueing theory, including several im-portant theorems and formulas. Advanced introduction to the field, but generallyavoids measure theory.

[3] E. Cınlar. Introduction to Stochastic Processes. Prentice-Hall, 1997.

Advanced introduction to stochastic processes. Develops the field using measure-theoretic notation, but without the measure theory proofs. Cıinlar is known for hiscareful use of notation, so readers would be well-served by aping his style. Book has aparticularly strong section on characterization of states in a Markov chain.

[4] K. L. B. Cook. Current Wideband MILSATCOM Infrastructure and The Futureof Bandwidth Availability. Aerospace and Electronic Systems Magazine, IEEE,25(12):23–28, 2010.

A descriptive paper on the structure of MILSATCOM, current limitations, andfuture avenues of expansion.

[5] A. Gilat. MATLAB: An Introduction with Applications. John Wiley & Sons, 2005.

A concise introduction to MATLAB programming. Particularly strong explana-tion of MATLAB scripts, functions, and plotting features.

[6] J. M. Harrison. Stochastic Networks and Activity Analysis. In Yu M. Suhov,editor, Analytic Methods in Applied Probability: In Memory of Fridrikh Karpelevich.American Mathematical Society, 2002.

Harrison conceptualizes “stochastic processing networks”. Multi-class queueingsystems are a special case of stochastic processing networks.

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[7] F. Kelly. Effective Bandwidths at Multi-class Queues. Queueing Systems, 9(1):5–15,1991.

Describes optimal control policy for a multi-class queue in which jobs can havevarious bandwidth requirements. Optimal policy is trunk reservation by effectivebandwidth.

[8] F. Kelly. Notes on Effective Bandwidths. In F. Kelly, S. Zachary, I. Ziedins, andI. Ziedins, editors, Stochastic Networks: Theory and Applications, Oxford SciencePublications, pages 141–68. Clarendon, 1996.

Introduction to the concept of “effective bandwidth” in queueing systems.

[9] F. Kelly and E. Yudovina. Stochastic Networks: Part III. 2011.

A set of lecture notes for a course on stochastic processes at the University ofCambridge. Includes topics, like truncation of state space, that are not generallytaught in an introductory setting.

[10] C. N. Laws. On Trunk Reservation in Loss Networks. In F. Kelly and R.J. Williams,editors, Stochastic networks, The IMA volumes in mathematics and its applications.Springer-Verlag, 1995.

A short paper that describes an optimal policy for a queueing system withclasses of jobs with different reward values.

[11] M. F. Neuts. Structured Stochastic Matrices of M/G/1 Type and Their Applications.Probability, pure and applied. Marcel Dekker, 1989.

High-level introduction to M/G/1 queues and their related stochastic matrices.Heavy use of measure theory, Laplace-Stieltjes transforms, and other theoretical tools.Lists 77 pages of references on queuing theory.

[12] C. H. Ng and B. H. Soong. Queueing Modelling Fundamentals with Applications inCommunication Networks. Wiley, 2008.

A well-written and concise introductory text on queueing theory. Discusses abroader range of material than Ross’s Introduction to Probability Models, with specialapplication to communication networks.

[13] J. A. Rice. Mathematical Statistics and Data Analysis. Number p. 3 in Duxburyadvanced series. Thomson/Brooks/Cole, 2007.

An introductory statistics text which does not shy away from mathematicalcontent. Useful description of goodness-of-fit tests and the χ2 distribution.

[14] P. Robert. Stochastic Networks and Queues. Stochastic Modelling and AppliedProbability. Springer, 2003.

An advanced book that treats M/M/1 and M/M/∞ queues with martingaletheory.

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[15] S. M. Ross. Introduction to Probability Models, Sixth Edition. Academic Press, 1997.

Presents background for queueing models. Non-measure theoretic, but neverthe-less encyclopedic in breadth. Go-to source for basic questions in queueing theory orstochastic processes.

[16] S. M. Ross. Simulation. Academic Press, 2002.

Concise yet comprehensive introduction to both the theory and techniques ofcomputer simulation. Useful chapters on discrete event simulation and statisticalvalidation.

[17] C. Sbarounis, R. Squires, T. Smigla, F. Faris, and A. Agarwal. Dynamic Bandwidthand Resource Allocation (DBRA) for MILSATCOM. In Military CommunicationsConference, 2004. MILCOM 2004. IEEE, volume 2, pages 758–764. IEEE, 2004.

Describes the advantages of statistical multiplexing in the MILSATCOM archi-tecture. However, proposed DBRA scheme is designed for packet-switched networks,not circuit-switched networks.

[18] M. Schwartz. Telecommunication Networks: Protocols, Modeling, and Analysis.Addison-Wesley series in electrical and computer engineering. Addison-Wesley Pub.Co., 1987.

A standard text in communication networks. Schwartz introduces circuit-switched andpacket-switched networks, and describes how queueing theory can be used to modelsuch networks.

[19] W. Wang and T. N. Saadawi. Trunk Congestion Control in Heterogeneous CircuitSwitched Networks. Communications, IEEE Transactions on, 40(7):1156–1161, 1992.

The authors use a Markov chain to describe a stochastic processing networkthat handles users with two different priority classes. Unlike most priority-basedsystems, this paper investigates a system in which preempted jobs do not return toservice.

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