calculating call blocking and utilization for...
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978-1-4577-0557-1/12/$26.00 ©2012 IEEE
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Calculating Call Blocking and Utilization for
Communication Satellites that Use Dynamic Resource
Allocation
Leah Rosenbaum Mohit Agrawal
Leah Birch Yacoub Kureh
Nam Lee UCLA Institute for Pure and Applied Mathematics
(IPAM) 460 Portola Plaza
Box 957121 Los Angeles, CA 90095-7121
James Hant Brian Wood
Eric Campbell James Gidney
The Aerospace Corporation 2310 E. El Segundo Blvd.
El Segundo, CA 90245 310-336-1388
Abstract— The performance of most satellite communication
(SATCOM) systems is characterized by loading analyses that
assess the percentage of users or total throughput a particular
system can satisfy. These analyses usually assume a static
allocation of resources in which users request communication
resources 100% of the time and higher priority users often
block lower priority users from getting service. However, the
loading of more dynamic, circuit networks such as the public-
switched telephone network (PSTN) is typically analyzed on a
statistical basis where the probability of a blocked call is
computed. These types of systems can potentially satisfy more
users than those that use static resource allocation because
they take advantage of statistical multiplexing. As SATCOM
moves toward a more dynamic concept of operations
(CONOPS) to take advantage of potential statistical
multiplexing gains, it is crucial to develop analysis capabilities
to evaluate performance.
In this paper, a method is developed to calculate call-blocking,
preemption, and resource utilization for dynamically-allocated
SATCOM systems in which users have different priorities and
bandwidth requirements. The first part of the study augments
the M/M/m queuing model to account for users with different
priorities and bandwidth requirements. In the second part of
the study, the model is used to predict the performance for two
competing traffic classes with different bandwidths or
priorities and highlight important trends. Finally, the third
part of the study directly compares the performance of static
and dynamic resource allocation approaches.
This work was performed by The Aerospace Corporation in
collaboration with a team of students representing the
Research in Industrial Projects for Students (RIPS) Program.
Administered by the UCLA Institute for Pure & Applied
Mathematics (IPAM), RIPS provides opportunities for high-
achieving undergraduate students to work in teams on real-
world research projects proposed by a sponsor from industry.
TABLE OF CONTENTS
1. INTRODUCTION ................................................. 1
2. THEORETICAL MODEL ..................................... 3
3. PERFORMANCE OF DYNAMIC RESOURCE
ALLOCATION ........................................................ 4
4. COMPARISON OF STATIC AND DYNAMIC
RESOURCE ALLOCATION ..................................... 7
5. CONCLUSIONS AND FUTURE WORK ................. 9
REFERENCES ......................................................... 9 BIOGRAPHIES ... ERROR! BOOKMARK NOT DEFINED.
1. INTRODUCTION
Satellite communication (SATCOM) systems often have
limited resources to satisfy communication circuits which
need to be managed among competing users who have
different priorities and bandwidth needs. Most of these
systems allocate resources on a static basis in which users
are given access to communication circuits for long periods
of time in priority order. A pictorial view of this type of
allocation approach is shown in Figure 1. This example
assumes a total system capacity of 100 Mbps and 18
requested circuits with different bandwidths and priorities
(high, medium, and low).
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Figure 1: Static Resource Allocation Approach
With this type of scheme, high-priority users are given their
own reserved channel, regardless of their usage pattern,
which causes lower priority users to be blocked and the
system to be underutilized. For this example, all low priority
users are blocked even though the server utilization is less
than 100 Mbps during most of the time.
A dynamic resource allocation approach is shown in Figure
2 in which users are allocated resources (in priority order)
only when those resources are specifically needed. For this
case, the system utilization is increased and more of the
lower priority users are satisfied even though some of these
users may be preempted by higher priority users.
Figure 2: Static Resource Allocation Approach
The potential benefit of implementing a dynamic resource
allocation scheme depends on the time varying bandwidth
needs for the different priority users. As the duty cycles of
the users decrease, a dynamic allocation approach can more
easily take advantage of multiplexing. In this paper,
classical queuing theory is expanded to highlight some of
the basic trade offs that determine the performance of static
vs. dynamic resource allocation schemes.
For this study, a SATCOM system with dynamic resource
allocation is modeled as the M/M/m/0 queuing model [1, 2]
shown in Figure 3 with m available circuits, no queuing
buffer, and circuit arrivals and departures described by
exponential distributions. This allows us to expand on
classical queuing theory to estimate performance and
determine trends.
Figure 3: M/M/m/0 Queuing Model
The following three types of traffic conditions are
considered for the dynamic resource allocation system:
1. Single traffic type: all requested circuits have the
same priority and bandwidth requirements
2. Two competing traffic classes with different
priorities
3. Two competing traffic classes with different
bandwidth requirements
Theoretical models for user satisfaction (or
blocking/preemption probability) and system utilization are
determined for these different traffic conditions and a direct
comparison is made between static and dynamic allocation
approaches.
The organization of this paper is as follows. In Section 2, a
theoretical model for dynamic resource allocation is
developed assuming a single traffic type or two competing
traffic classes with different priorities or bandwidths. In
Section 3, the theoretical model is used to generate results
that demonstrate some of the basic performance trends.
µ
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In Section 4, the performance of static and dynamic
allocation schemes is compared for two competing traffic
classes with different priorities or different bandwidths.
Finally, conclusions and suggestions for future work are
presented in Section 5.
2. THEORETICAL MODEL
In this section, theoretical models for dynamic resource
allocation are developed for a single traffic type, two
competing priorities, and two competing bandwidths. To
evaluate system performance, we consider the following
two performance measures: call blocking/preemption
probability and server (or bandwidth) utilization. A call is
blocked when there are not enough servers available in the
system to handle the job. Preemption occurs when a low
priority user gets kicked off a server by a high-priority user
who requests to use the system. Server utilization describes
the average system utilization or how much bandwidth is
being occupied on an average basis. These measures are
tracked as a function of the traffic intensity, ρ, which is the
ratio of overall arrival and departure rates from the system.
A discrete-event simulation model [4] was also generated in
MATLAB [3] to verify all theoretical results.
Single Traffic Type
For one job type, we can consider a stochastic processing
network with m servers and a queue of length 0. As such,
there can be at most m jobs in the system at any time. If a
job seeks to enter the system but no free servers are
available, the job is blocked. Because blocked jobs are lost
forever, this system is known as the Erlang loss system [2].
A state transition diagram for this type of system is shown
in Figure 4.
Figure 4: State Transition Diagram for an M/M/m/0
Queuing System
System state is defined as the number of servers that are
currently occupied and it changes whenever a new job
arrives or leaves the system. The inter-arrival times of jobs
entering the system is described by an exponential
distribution with an average arrival rate of The time
required to service each job is also described by an
exponential distribution with an average service rate of
Once the system is in a given state, the probability of
entering another state is fixed and independent of the
system’s past states and thus the system can be modeled by
a finite state, continuous time Markov process [1]. When the
underlying operating policy is first-in-first-out (FIFO) the
M/M/m/0 system can be described by the following
infinitesimal transition rate matrix, Λ.
[ ]
(1)
Assuming the process is stationary and irreducible, the
probability that the system is in a particular state, π, is
calculated by finding the unique solution to the following
two equations:
(2)
(3)
Blocking probability can then be calculated as the
probability that the system is in state m (that is, that the
server is fully utilized) and the mean system occupancy (or
server utilization) can be calculated as a weighted sum of
the probabilities of being in each state.
A similar analysis can be done for a dynamic allocation
system with two competing priorities, however, now the
state transition diagram and infinitesimal transition rate
matrix will be different. To gain some insight into how to
develop a state transition diagram for competing priorities,
consider the m=1 case (shown in Figure 5).
Figure 5: Continuous-Time Markov Chain for
Competing Priorities
An m=1 system with two competing priorities can be in one
of 3 states: unoccupied (‘0’), servicing a low-priority circuit
(‘low’), or servicing a high-priority circuit (‘high’). State
transitions are determined by the arrival rate of high (H) or
low (L) priority circuits and the service rate for those
circuits (The infinitesimal transition rate matrix, Λ, for
this system is given by equation 4.
highlow0
H
H
L
4
[
] (4)
This model can be extended to any value of m and the
blocking probability and server utilization can be calculated
based on equations (2) and (3) for different values of H, L,
and . These values can be used to determine the
corresponding traffic intensities for high and low priority
traffic defined as H = (H/and L = (L/respectively.
A continuous time Markov chain can also be defined for a
dynamic allocation system with two competing bandwidths.
To better understand how this is done, consider a system
with a server capacity of 4 bandwidth units and two job
classes: jobs requiring 1 bandwidth unit (with an arrival rate
of λ1) and jobs requiring 2 bandwidth units (with an arrival
rate of λ2). Assuming both traffic classes have the same
service rate, , the Markov chain shown in Figure 6 can be
used to describe the system.
Figure 6: Continuous-Time Markov chain for
Competing Bandwidths
Note that this chain is two dimensional and the ordered pair
(n1 , n2 ) indicates the state of the system having n1 jobs of
the first class (those requesting 1 bandwidth unit) and n2
jobs of the second class (those requesting 2 bandwidth
units). An infinitesimal transition rate matrix can be
generated for this Markov chain or for any two competing
bandwidths with an arbitrary number of servers. The
blocking probability and server utilization can then be
calculated as a function of the total traffic intensity and the
ratio of arrival rates for both classes. For the case of
competing bandwidth classes, the total traffic intensity is
given by equation (5) where λi is the arrival rate for
bandwidth class i, Bi is number of servers (or bandwidth
units) requested for class i, m is the total number of servers
(or bandwidth units), and is the service rate for each user.
∑
(5)
3. PERFORMANCE OF DYNAMIC RESOURCE
ALLOCATION
In this section, the theoretical models developed in Section
2 are used to highlight some of the basic trends of a
dynamic resource allocation system with a single traffic
type, two competing priorities, and two competing
bandwidth classes.
Single Traffic Type
Figure 7 shows the performance of a dynamic resource
allocation system with a single traffic type. Blocking
probability and mean server utilization is plotted in (a) and
(b), respectively, for different numbers of servers, m. As
expected, both blocking probability and server utilization
increase with increasing traffic intensity, however,
performance is much better for larger m. This implies that
systems with a larger number of servers (greater bandwidth
resolution) have less blocking and can more efficiently use
resources.
Figure 7: Performance for a Single Traffic Type
(a)
(b)
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Two Competing Priorities
Assuming a queuing system with 100 servers (m=100), the
performance of two competing priorities (high and low) is
plotted in Figure 8 and Figure 9, respectively. Figure 8 (a)
and (b) plot the blocking probability and server utilization
for high priority traffic as a function of the high priority
traffic intensity, H, for different arrival ratios of high and
low priority traffic (H/L). The performance of an
equivalent single-traffic type system (M/M/100) is
superimposed on the plots to assess the effect of
prioritization. Results show that high priority traffic only
competes with itself and its performance is completely
determined by the high priority traffic intensity. Regardless
of the ratio of arrivals of high and low priority traffic, high
priority traffic performs identically to an M/M/100 system
at the high priority traffic intensity.
Figure 8: Performance for Two Competing Priorities
(High Priority Users)
Figure 9 (a) and (b) plot the blocking probability and server
utilization for low priority traffic as a function of the total
traffic intensity for different arrival ratios of high and low
priority traffic (H/L). The performance of an equivalent
single-traffic type system (M/M/100) is again superimposed
on the plots to assess the effect of prioritization. Results
show that the performance of low priority traffic, on the
other hand, is highly dependent on the ratio of high and low
priority traffic. When most of the traffic arrivals are low
priority (H/L is small) than the system behaves similarly to
an M/M/100 system. However, when most of the traffic
arrivals are high priority (H/L is large) than the low
priority traffic is preempted and its performance degrades
considerably from M/M/100.
Figure 9: Performance for Two Competing Priorities
(Low Priority Users)
Two Competing Bandwidth Classes
The results for a system with 100 total servers and two
competing traffic classes with bandwidths of 1 and 10
servers (or bandwidth units) each are shown in Figure 10
below. Blocking probability and mean server utilization are
plotted as a function of traffic intensity with the ratio of
arrivals for both traffic classes (1/1010) as a parameter.
Note that the ratio of arrivals is weighted by the bandwidth
required for each traffic class. To gain greater insight,
results are plotted with cases in which each traffic class is
by itself. This corresponds to an M/M/100 model for the
(a)
(b)
(a)
(b)
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bandwidth 1 class and an M/M/10 model for the bandwidth
10 class.
Figure 10: Total Performance for Two Competing
Bandwidths
Results show total blocking probability is lower-bounded
and server utilization is upper-bounded by the smallest
bandwidth class (M/M/100). Interestingly, the blocking
probability is not upper-bounded (and the server utilization
is not lower bounded) by the largest bandwidth class
(M/M/10). When the ratio of small and large bandwidth
traffic is near 1, the system actually has worse performance
than if the large bandwidth traffic was by itself. This is
because the small bandwidth traffic can more easily block
the large bandwidth traffic from getting service.
The blocking probability of the small and large bandwidth
users is shown in Figure 11 (a) and (b), respectively.
Blocking probability is plotted as a function of the ratio of
arrivals for the large and small bandwidth traffic classes
(1/1010) with total traffic intensity as a parameter. To gain
greater insight into how the two traffic classes compete for
resources, the performance of equivalent single-traffic type
systems (M/M/100 for the bandwidth 1 class and M/M/10
for the bandwidth 10 class) are superimposed onto the
results.
Figure 11: Performance of Small and Large Bandwidth
Users for Two Competing Bandwidths
Figure 11 (a) shows that when a majority of the traffic is
large bandwidth users (1/1010 is small) the blocking
probability for small bandwidth users is similar the case
where only the large bandwidth traffic is competing with
itself (M/M/10). Conversely, when most of the traffic
consists of small bandwidth users (1/1010 is large) than
blocking probability is similar to the case where only the
small-bandwidth traffic is competing with itself (M/M/100).
Interestingly, small bandwidth users have the best
performance when there is nearly an equal ratio of large and
small bandwidth users. This is because small bandwidth
users can more easily take advantage of available servers
and block the large bandwidth users from getting service.
Figure 11 (b) shows a more consistent trend for the large
bandwidth users. When a majority of the traffic is large
bandwidth users (1/1010 is small), the performance of
large bandwidth users is dominated by these users
competing with themselves (M/M/10). As the amount of
small bandwidth traffic increases, more of the large
bandwidth users begin to get blocked by the small
bandwidth users. At the point where the ratio of small-to-
(a)
(b)
(a)
(b)
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large bandwidth users is nearly equal (1/1010=1), almost
no large bandwidth users are able to get through at the
higher traffic intensities (=1.5 or 2).
4. COMPARISON OF STATIC AND DYNAMIC
RESOURCE ALLOCATION
With an analysis method developed to compute the blocking
probability and server utilization for dynamic SATCOM
systems with competing priorities and bandwidths, the
performance of static and dynamic allocation can be
compared. Figure 12 compares the total performance of
static and dynamic resource allocation for a SATCOM
system with two competing priorities (high and low) and
100 total servers. The arrival rates of the low and high
priority traffic are equal. A static system must pre-allocate
resources for each priority and the following combinations
of high:low priority servers were tested: 99:1, 95:5, 90:10,
80:20, 70:30, 60:40, 50:50. The dynamic system was able
to allocate resources dynamically, with low priority circuits
being preempted by high priority circuits when all servers
were occupied. Total satisfaction (defined as 1 – the
blocking/preemption probability) was plotted vs. server
utilization for all the cases tested.
Figure 12: Comparison of Static and Dynamic Resource
Allocation for Two Competing Priorities (Total
Performance)
Results show dynamic resource allocation outperforms
static resource allocation for all possible configurations.
Regardless of the number of servers pre-allocated to high or
low priority users by a static allocation scheme, dynamic
resource allocation always achieves better user satisfaction
at a given server utilization.
The performance of high and low priority traffic classes
under static and dynamic resource allocation is shown in
Figure 13. User satisfaction for high and low priority users
is plotted as a function of traffic intensity in Figure 13 (a)
and (b), respectively, for all system configurations
considered. Results show that for the high priority users,
dynamic resource allocation always outperforms static
resource allocation. However, that is not always true for the
low priority users. At lower traffic intensities, dynamic
allocation provides improved performance for low priority
traffic. At higher traffic intensities, however, dynamic
allocation allows high priority requests to outcompete low
priority requests, leading to lower satisfaction than static
allocation cases where resources have been set aside for low
priority users.
Figure 13: Comparison of Static and Dynamic Resource
Allocation for Two Competing Priorities (Performance
for each priority)
A similar comparison of static and dynamic allocation
schemes was generated for two competing bandwidth
classes. Figure 14 compares the performance of static and
dynamic resource allocation for a SATCOM system with
100 total servers and two competing traffic classes with
different bandwidth requirements: one that requests 1 server
and the other that requests 10 servers. The arrival rates of
the small and large bandwidth classes were assumed to be
equal (1/1010=1). The static system pre-allocated 50
servers for each bandwidth class while the dynamic system
was able to allocate resources for both classes dynamically.
Total satisfaction (defined as 1 – the blocking probability)
(a)
(b)
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and server utilization were plotted vs. traffic intensity in
Figure 14 (a) and (b), respectively.
Figure 14: Comparison of Static and Dynamic Resource
Allocation for Two Competing Bandwidths (Total
Performance)
Results show that the total performance of the dynamic
resource allocation system has slightly better performance
than the static system, with higher user satisfaction and
server utilization at all traffic intensities. The performance
for small and large bandwidth users is shown in Figure 15
and Figure 16, respectively.
Figure 15 plots user satisfaction and server utilization for
small bandwidth users as a function traffic intensity for the
both the dynamic allocation system (shown in green) and
the static allocation system (shown in blue).
Figure 15: Comparison of Static and Dynamic Resource
Allocation for Two Competing Bandwidths (Small
Bandwidth Users)
Results show that dynamic resource allocation has much
better performance at the higher traffic intensities. As traffic
intensity approaches 1, dynamic resource allocation allows
small bandwidth users to utilize more than 50% of the
bandwidth resources, resulting in greater user satisfaction
and server utilization. The static resource allocation system
only allocates 50 servers to these users which greatly
reduces their satisfaction and utilization at the higher traffic
intensities.
This improvement in performance for small bandwidth users
at high traffic intensities comes at the price of the large
bandwidth users (as shown in Figure 16). Figure 16 plots
user satisfaction and server utilization for large bandwidth
users as a function of traffic intensity for both the dynamic
allocation system (shown in green) and the static allocation
system (shown in blue).
(a)
(b)
(a)
(b)
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Figure 16: Comparison of Static and Dynamic Resource
Allocation for Two Competing Bandwidths (Large
Bandwidth Users)
Results show that at the higher traffic intensities, the
performance of the large bandwidth users is degraded
considerably by the dynamic resource allocation scheme.
This is because at the higher traffic intensities, dynamic
resource allocation allows the small bandwidth users to take
away resources from the large bandwidth users, reducing
their satisfaction and utilization. The static allocation
system, on the other hand, pre-allocates 50 servers for just
the large bandwidth users, resulting in much better
performance. Since large bandwidth users are so
disadvantaged, it may necessary to fence off resources for
them if dynamic allocation systems are finally implemented
in a SATCOM system.
5. CONCLUSIONS AND FUTURE WORK
An analytical model has been generated to measure user
satisfaction (or blocking/preemption probability) and
resource utilization for dynamically-allocated SATCOM
systems that have users with competing priorities and
bandwidths. Results show that users who request a smaller
fraction of the total bandwidth resources have better
performance (less blocking and higher utilization) than
higher bandwidth users. For competing priorities, high
priority traffic only competes with itself and its performance
is determined by the high priority traffic intensity. Lower
priority users are highly dependent on the amount of high-
priority traffic and as the ratio of high priority jobs
increases, more low priority jobs are preempted and their
bandwidth utilization is degraded. For competing
bandwidths, total performance is upper bounded by the
smaller bandwidth traffic class and small bandwidth jobs
can more easily block large bandwidth jobs from getting
service. Systems with large and small bandwidth jobs
arriving at similar rates perform marginally worse than if the
large bandwidth jobs were only competing with themselves.
Dynamic allocation schemes provide better overall
performance than comparable static allocation schemes,
however, at the higher traffic intensities they provide worse
performance for low priority and large bandwidth users.
Future work will consider more sophisticated models of
satellite resources and traffic load, including the effects of
frequency channels, time-slots, antenna coverage, beam
pointing, and requested circuits with duty cycles. Additional
model enhancements may also include allowing for more
than two competing priority and bandwidth classes, reentry
procedures for preempted jobs, and improvement to the
dynamic allocation algorithm. It will also be important to
model some of the real-world consequences of
implementing a Demand Assigned Multiple Access
(DAMA) scheme for SATCOM including the effects of
protocol delays and network management.
REFERENCES
[1] S. M. Ross. Introduction to Probability Models, Sixth
Edition. Academic Press, 1997
[2] E. Cınlar. Introduction to Stochastic Processes. Prentice-
Hall, 1997.
[3] A. Gilat. MATLAB: An Introduction with Applications.
John Wiley & Sons, 2005.
[4] S. M. Ross. Simulation. Academic Press, 2002.
(a)
(b)