1

Queueing with Heterogeneous Users: BlockProbability and Sojourn times

Veeraruna Kavitha and Raman Kumar SinhaIEOR, Indian Institute of Technology Bombay, India

Abstract—We consider a queueing system with heterogeneousagents. One class of agents demand immediate service, and wouldleave the system if not provided. The agents of the second classhave longer job requirements and can wait for their turn. Wediscuss the achievable region of such a two class system, whichis the set of all possible pairs of performance metrics. Blockingprobability is the relevant performance metric for impatientclass while the expected sojourn time is appropriate for thesecond tolerant class. We consider static (time invariant and stateindependent) and dynamic (state dependent) scheduling policies.In queueing systems with homogeneous agents, where expectedsojourn time is the performance metric for both the classes, weshow that the static and dynamic achievable regions coincide.However, this is not the case with heterogeneous setting. Weobtain the achievable region, under static policies. We consideran example dynamic policy and show that the dynamic achievableregion is strictly bigger than the static region.

We conjecture a pseudo conservation law, in a fluid limit forimpatient customers, which relates the blocking probability ofeager customers with the expected sojourn time of the tolerantcustomers. We provide a partial proof and use the pseudoconservation law to obtain the static achievable region. Wevalidate the pseudo conservation law using two example familiesof static schedulers, both of which achieve all the points onthe achievable region. Along the way we obtain smooth control(sharing) of resources between voice and data calls.

Index terms– Heterogeneous agents, achievable region, processorsharing, resource sharing, dynamic and static scheduling.

I. INTRODUCTIONWe consider queueing systems with heterogeneous classes.

First one is a class of impatient agents. They reject the systemif service is not offered almost immediately. One would requirea parallel (upto K servers) service-offer facility to handlethis class. Blocking probability, the probability that an agentreturns without service, is important metric for this class. Theagents of the second class are tolerant, can wait for their turn.However, their satisfaction depends upon the expected sojourntime (total time spent in the system).

An achievable region for an n-class system is the set ofrelevant performance vectors (pm1, · · · , pmn), obtained byvarying all possible scheduling policies (e.g., [12], [15], [16]).In heterogeneous setting, the achievable region is the set ofpairs of blocking probability and expected sojourn times.

The achievable region is well understood for homogeneousclasses, when the performance metric of both the classes isexpected sojourn time (e.g., conservation laws, pioneered by[9]). Coffman and Mitrani [11] were the first to identify suchachievable regions. Multi-class single server queueing systemspose nice geometric structure (polytopes) for achievable region(e.g., [11], [12]). The scheduling policies are dynamic if they

depend upon the (time varying) system state. Static policiesare time invariant and state independent. Our first observationis that the homogeneous achievable region is the same understatic as well as dynamic policies.

In this paper we are studying the achievable region ofqueueing systems with heterogeneous classes.

A parametrized family of scheduling policies is calledcomplete by [20], if it achieves all possible performancevectors of the achievable region, average waiting times intheir context. A complete scheduling class can be used tofind the optimal control policy over all scheduling disciplines.Discriminatory processor sharing (DPS) class of parametrizeddynamic priority schedulers is identified as a complete familyin case of two class M/G/1 queue in [21]. Many morefamilies of scheduling policies are identified to be complete.

Our contribution

We conjecture a relation between the expected sojourn timeand the blocking probability, which would be valid for anystatic scheduling policy, and, call it a pseudo conservationlaw. This pseudo conservation law is valid in a fluid limit forshort job impatient agents and we provide a partial proof. Wethen show that two sets of scheduling (PS and CD) policiessatisfy this conservation law and also achieve all the pointsof the resulting achievable region. In both the policies, theadmission of an impatient agent disrupts the service of theongoing tolerant agent (if any). In the first case the entiresystem capacity is transferred to the impatient agent, whilea fixed fraction of capacity is transferred in the second case.We refer the first system as PS policy, as here the impatientagents derive service in the well known processor sharingmode. At maximum K impatient agents can share the facility.In the second policy, the service of the tolerant agent iscontinued with the remaining capacity. Admission of anotherimpatient agent results in the transfer of an additional fractionof capacity, while a departure results in the transfer back ofthe same fraction. The second policy is referred as capacitydivision-CD policy, because the capacity is divided betweenthe two classes. The tolerant agents are always served in serialfashion, i.e., one at a time in both the policies.

We solve an appropriate set of balance equations to obtainthe busy probability. We majorly use a domination techniqueto study the tolerant class. The actual system is sandwiched inbetween the two M/G/1 queues. The difference in the sojourntime performance between the two queues converges to zeroas arrival-departure rates of the impatient agents converge to

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infinity, while maintaining their ratio fixed. Typically agentswith long job requirements form tolerant class, while the im-patient ones demand for short jobs (e.g., super markets, data-voice calls etc.). Thus they operate in the precise asymptoticregime, for which our results are accurate.

ApplicationsThe problem of resource allocation, between data-voice

calls of a communication network is a loosely related topic(e.g., [5], [6]). Voice calls are dropped if server is not available,while data calls can wait. The voice, data calls respectivelyform impatient and tolerant agents. In [5] the authors considera three channel pool scheme to obtain a novel adjustableboundary based channel allocation scheme with pre-emptivepriority for integrated data-voice networks. They attain variouslevels of priority by adjusting the division of total availablechannels among the three pools. In [6] authors again considerchannel allocation scheme for packet level allocations. Thesepapers discuss coarse sharing of resources between data-voicecalls. While we provide a smooth control: by varying theadmission parameter p (continuously) in the interval [0, 1], onecan achieve any pair of performance metrics for data-voicecalls on the static achievable region.

Various applications (e.g., computers, communication net-works, manufacturing systems) can be modelled as multi-classqueueing systems and dynamic control of such systems isan important aspect. One of the main techniques for suchproblems is to characterize the achievable region, then useoptimization methods to obtain optimal control policy (forexample [17], [18], [19]). Once the heterogeneous achievableregion is known, many relevant optimization problems can besolved in a similar way. If further a good complete family isidentified, the optimization problem gets simplified greatly.For example, our CD/PS policies are parametrized by K,the number of parallel service facilities and p the probabilitywith which an impatient job is admitted. Thus the problemof finding the optimal expected sojourn time of data callsin a communication network, given a constraint on blockingprobability of voice calls can readily be obtained, by optimallychoosing K, p.

In a super market the long jobs wait, while short jobs areprovided fast service via dedicated express counters. Alterna-tively, if one use the same counters to serve both the jobs andif selected (controlled) short jobs pre-empt the long jobs, onecould obtain optimal design using our static achievable region.For example, one can obtain the optimal fraction of short jobslost, given a constraint on the expected sojourn time of longjobs.

Consider a communication system with K orthogonal chan-nels. Initially all the channels are dedicated to data calls. Asand when the voice calls arrive, one by one the channels aretransferred and data calls use the remaining. Our CD policycaptures this scenario precisely. If the voice calls are servedat the highest possible rate as with PS policy, it improvesthe chances of a free server being available to subsequentvoice arrivals. The two achievable regions overlap, but PShas a bigger region (when number of maximum parallel callsis fixed) as it attains a smaller blocking probability.

Literature survey, related to Queueing systems

Our paper mainly deals with resource sharing betweencustomers who are willing to wait for the service and thecustomers who are not willing to wait much. To the best ofour knowledge we are not aware of a work that directly studiesthis type of a heterogeneous achievable region. Some variantsof queueing systems (e.g., [23], [13], [14], [7], [4], [10], [2]etc) have some connections to few parts of our models andwe brief such papers here.

In [13] authors consider multi-class queueing system witheager and tolerant customers. This is the work that is closestto our work, especially to CD policy. The tolerant customersutilize all the remaining servers (hence are work conserving)in our model, while [13] considers tolerant customers also inmulti-server mode. If there is only one tolerant customer andif say n > 1 free servers are available, the tolerant customeruses only one server in [13], while with our CD policy thetolerant user is served using the consolidated capacity of allthe n servers. The authors in [13] obtain a set of (balance)equations, solving which stationary probabilities (and then thestationary performance) can be derived. We provide a closedform expression for these performance measures in fluid limitfor eager customers. We further discuss a ‘policy-independent’pseudo conservation law.

In [2] authors analyzed an M/G/1 queue with Poissoninterruptions, caused either by server breakdown or by thearrival of higher priority agents. This system is similar to ourPS/CD policy with single server.

In [23], the authors consider time limited autonomouspolling system. Here the server visits a finite number of queuesin a periodic manner, while spending a random (exponentiallydistributed) visit time at each queue, independent of thestatus of the queues. It spends the designated exponentialtime, even if the queue gets/is empty. The tolerant agents ofour PS model can be studied using their results. Howevertheir expressions are complicated, while we derive simpleexpressions in an appropriate asymptotic limit. Further, similartechniques are used to study the CD model.

In [14] Sleptchenko et. al considered a multi-class M/M/kqueueing system with two priority groups, both groups ofcustomers are tolerant and each priority group may consistof multiple customer types. Different customers type havingtheir own arrival and service rates. Upon arrival a high prioritycustomer pre-empts the lower one, if all the servers arebusy and some are serving low priority customers. The maincontribution is that they obtain the stationary state probabil-ities, which provides many more performance measure (e.g.,moments of type wise waiting customers, mean number oflow-priority customers interrupted etc.) beyond the stationarymean values. As already mentioned, this paper only considerstolerant customers.

In [7], authors consider multi-class single-server queuewith K classes of customers. Arrival of each class followsindependent Poisson process and the service time is generallydistributed. The service capacity is shared simultaneouslyamong all customers present in proportion to the respectiveclass-dependent weights. They derived closed form approx-

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imation for the mean conditional and unconditional sojourntime of each classes and verified it in light traffic and hightraffic modes. In [4], authors considered M/M/K queue withm > 2 number of priority classes. They reduce the m-dimensional Markov chain to 1-dimension Markov chain usingthe busy period of high priority customers. They modelledthe one dimensional Markov chain as QBD (Quasi BirthDeath) processes and calculate the mean sojourn time for lowpriorities customers. Both these papers only deal with tolerantcustomers.

In section II we describe the problem. We conjecture thecomplete static region and provide a partial proof in sectionIII. PS and CD models are respectively analyzed in sectionsIV and V and are compared in section V. Using an exampledynamic policy for PS model we showed in section VII thatthe heterogeneous dynamic achievable region is strictly biggerthan the static region.

II. PROBLEM STATEMENT AND SYSTEM MODEL

We refer the impatient/eager customers by �-customerswhile the tolerant customers are referred to as τ -customers.The system has a fixed server capacity, that needs to be sharedbetween the two classes of customers. The exact sharing ofcapacity depends upon the allocation/scheduling policy. Forexample the system can serve K customers in parallel forsome K, by dividing the server capacity among the customersunder service. The system can chose to vary K dynamically,e.g., processor sharing. The system can chose to serve onecustomer with full capacity etc. In this paper we discuss twoexample sets of scheduling policies. We only consider ‘τ -workconserving’ policies, wherein the τ -customers utilize all theremaining server capacity.

A. Arrival process and the Jobs

The arrival processes are modelled by independent Poissonprocesses, with rates λ� and λτ respectively. The job require-ments for both the classes are exponentially distributed. Thetime required to complete a job, depends upon the schedulingpolicy. If a τ -customer (�-customer) is served with full servercapacity, then the service time is exponentially distributed withparameter µτ (respectively µ�).

B. Achievable region

The two classes of users have different goals and hencenaturally require different qualities of service (QoS). An eager�-agent would leave the facility without service, if serviceis not provided almost immediately. Block probability PB ,the probability of such an event is an important performancemetric for �-class. The service of a typical τ -customer canpossibly be interrupted, possibly to provide required QoSfor �-agents, and a typical τ -agent can face several suchinterruptions during its service. Thus the expected sojourn timeE[Sτ ], the expected value of the total time spent by a typicalagent would be an appropriate QoS for τ -class. Either of theseperformance metrics depend upon the scheduler β used.

A scheduling decision is required at every �-arrival in-stance1, because the �-agents are impatient. Because of thisthe service of a τ -agent can be pre-empted several times. Thusthe appropriate QoS for τ -agents is the expected sojourn time,and not the expected waiting time. Therefore the achievableregion is given by:

Ahetero = {(PB(β), E[Sτ (β)]) : β is a scheduler}.

In this paper we consider (τ ) static policies, wherein the �-admission rules do not depend upon the status of the τ -class.The probability of admission, p, is an important parameter ofany such scheduling policy. Further, the (maximum) numberof �-calls served in parallel and the sharing of resourcesbetween � and τ customers is also a part of the schedulingdecision. For example, the system may allocate/transfer entireserver capacity to the first admitted �-arrival. It may processor-share the capacity among the further admitted �-customers.There may be a limit on the number of �-customers that cansimultaneously share the capacity. Alternatively the systemmay allocate a fixed fraction of the server capacity to eachadmitted �-arrival and the remaining is allocated to τ -classetc. All these rules are independent of the τ -state (e.g., thenumber of τ customers in the system, waiting time of themetc). This implies that �-calls pre-empt τ -call when required.In all a τ -static policy implies that an �-arrival is admittedwith some probability p, and further admission also dependsupon the number of �-calls already in the system, but not onτ -state. Mathematically a static achievable region is defined:

Astatichetero ={(PB(β

CSp ), E[Sτ (β

CSp )]

):

0 ≤ p ≤ 1 and (CS) a capacity sharing rule}. (1)

We primarily analyze the static achievable region. Towards theend, in section VII an example dynamic policy is consideredto show that the achievable region with dynamic policies isstrictly bigger than the static achievable region.

C. Short-Frequent Job (SFJ) limits

The �-class has short job requirements. If one considerslimit µ� →∞, the impact of �-customers becomes negligibleat the limit. To obtain a more general and useful result, wealso increase the �-arrival rate while µ� → ∞. That is, every�-agent may utilize the server for a short duration, but thesystem has to attend the �-agents frequently. Because of this�-agents cause significant impact even in the limit. To be moreprecise we consider the limits µ� → ∞ and λ� → ∞ whilethe load factor ρ� = λ�/µ� is maintained constant. We referthis as “Short-Frequent Job (SFJ) limits”.

III. ENTIRE STATIC REGION: PSEUDO CONSERVATIONLAW

In a (homogeneous) multi-class queueing system, with alltolerant classes, a work conservation law holds. The total

1On the contrary, in homogeneous setting two or more classes of agentswait at their waiting lines and scheduling epochs are the service comple-tion/departure epochs. The scheduler had to decide which class to be servednext. While in heterogeneous setting, at any departure epoch there is only oneclass of agents possibly waiting and hence no decision is required.

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workload in the system remains the same irrespective of thescheduling policy, as long as the server does not idle duringbusy period. Further by Little’s law and Wald’s lemma, alinear combination of expected sojourn (or waiting) time ofdifferent classes of customers remains the same irrespectiveof the scheduling policy (e.g., [12]).

The above is obviously true when the incoming workloadremains the same. However in our heterogeneous setting, the�-customers depart the system, if service is not offered almostimmediately. And this depends upon the scheduling policy.Thus the workload arriving into the system itself changeswith different scheduling policies and naturally one may notexpect work conservation. However if the amount of workblocked remains the same, one can anticipate a different kindof work conservation. We conjecture that given a probabilityof blocking, irrespective of the way the �-agents are blockedand irrespective of the way the τ -agents are served, the τ -expected sojourn time remains the same2. And this could beconjectured only in SFJ limit and when the policies do notdepend upon the τ -state.

In SFJ limit, �-agents will have fluid arrivals and departures.Given the �-load factor (ρ�) and the probability of blocking(pB), in the SFJ limit, the �-agents occupy ρ�(1−pB) fractionof system resources at all the times. Hence we conjecturethat the τ - performance equals that of an M/M/1 queue withsmaller service rate µτ (1−ρ�(1−pB)), and that the expectedsojourn time for any 0 ≤ pB ≤ 1 equals:

ESτ (pB) :=1

µτ (1− ρ�(1− pB))− λτif ρ�(1− pB) + ρτ < 1.

(2)

Conjecture: Static achievable region, in SFJ limit, equals:

Aheterostatic ={(pB , ESτ (pB)

) ∣∣∣ pB ∈ [0, 1], ρ�(1− pB) + ρτ < 1}.We would like to refer the equation (2) as a pseudo conser-

vation law, as it provides the expected sojourn time in termsof the fraction blocked (lost). This would require an explicitproof which is provided immediately.

A. Proof of Pseudo Conservation Law

We will prove Pseudo conservation law (Theorem 1 givenbelow) under some of the following assumptions:A.1) The schedulers are τ -static (do not depend upon τ -state).A.2) The scheduling policies depend only upon the number

of �-customers already in the system.A.3) The τ -customers are served in serial fashion. The sched-

uler is work-conserving for τ -customers, i.e., it servesthe τ -customers with all the left over server capacity, ifthere are any.

We begin with some discussions, which form a part of theproof.

We obtain different �-systems with different µ� using thefollowing special construction (without loss of generality).First we consider a sequence of � inter-arrival, job-requirementsequences {A1�,n, B1�,n}n for µ� = 1. Note that E[A1�,n] =

2With one class of customers, the expected sojourn time by Little’s lawand Wald’s lemma is proportional to the workload in the system

1/ρ� and E[B1�,n] = 1 for all n. Then obtain the sequence{Aµ��,n, Bµ��,n}n for any general µ�, by multiplying the corre-sponding arrival-departure sequences by 1/µ�, i.e.,

Aµ��,n =A1�,nµ�

and Bµ��,n =B1�,nµ�

for every n, µ�. (3)

Since the decisions are independent of τ -customers (byA.1) one can identify a renewal process, corresponding onlyto the �-system, alternating between �-busy periods and �-idleperiods, both of which depend only upon �-customers. Let Xa,Xtot respectively represent the number of �-customers thatreceived service and the number that arrived in one renewalcycle. By renewal reward theorem (RRT) applied twice weobtain the blocking probability:

(1− PB) =E[Xa]

E[Xtot]. (4)

By A.1, we are considering τ -static scheduling policies here,wherein the decision do not depend upon the τ -state of thesystem. By A.2 the policies depend only upon the numberof �-customers in the system. Thus, if Xµ(t) represents thenumber of �-customers in the system at time t with µ� = µ,then

Xµ(t) = X1(µt) for any µ. (5)

Hence with A.1-2, PB remains the same for all µ� (see (4)).The length of a typical renewal cycle equals the arrival in-

stance of Xtot-th customer, which by the special constructionequals:

Xtot∑n=1

Aµ��,n =1

µ�

Xtot∑n=1

A1�,n,

By Wald’s Lemma the expected value of the length of arenewal cycle (with µ�) equals E[Xtot]/λ�. The total amountof time for which �-customers utilize the server during onetypical renewal cycle, irrespective of the way the service isoffered and the way the customers are admitted, stochasticallyequals

Xa∑n=1

Bµ��,n,

whose expected value clearly equals E[Xa]/µ�. Thus the longrun fraction of server time available for τ customers by RRTequals:

ντ (µ�) = ντ (PB) =E[Xtot]/λ� − E[Xa]/µ�

E[Xtot]/λ�

= 1− ρ�E[Xa]

E[Xtot]= 1− ρ�(1− PB).

This is true for any µ� and for any static scheduler. Hence,we have the following:

Theorem 1: a) Assume A.1. For any µ� and for any staticscheduler, the long run fraction of server capacity availableto τ -customers depends only upon ρ� and the probability ofblocking PB of � customers:

ντ (µ�) = ντ (PB(µ�)) = 1− ρ�(1− PB).

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b) Assume A.1-2. Then PB and hence ντ remains the samefor all µ�; andc) When µ� →∞ the accumulated amount of server capacityavailable to τ -customers at time t, Rµ�(t), converges to aconstant curve with value µτντ . This convergence is uniformover any bounded time intervals and almost surely:

supt∈[0,W ]

∣∣Rµ�(t)− ντ t∣∣→ 0 almost surely, as µ� →∞.for any 0 < W λτ .

Proof: Parts (a) and (b) are already proved. Parts (c)-(d) willbe proved below.

We again consider the special construction given by (3) and(5) is true by A.1-2. Let Rµ(t) represent the total amount ofresidual server time available for τ -customers till time t. ByRRT, as depicted by part (a)

Rµ(t)

t→ ντ (µ) a.s. as t→∞ for any µ.

By part (b), ντ (µ) = ντ is the same for all µ. Now, clearly

Rµ(t)

t=R1(µt)

µtwhich implies Rµ(t) =

R1(µt)

µ.

By Theorem 4 of Appendix A, a function version of RRT, wehave:

supt∈[0,W ]

∣∣∣R1(µt)µ

− ντ t∣∣∣→ 0 almost surely, as µ→∞.

for any 0 < W

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IV. PROCESSOR SHARING PS − (p,K) SCHEDULERS

Any �-arrival is admitted to the system with probability p,independent of τ -state. Once admitted it will pre-empt theexisting τ -agent, if any. We consider K-processor sharingservice discipline for �-agents. If there is only one agent of the�-class receiving service, it is served with maximum capacity,i.e., using capacity µ�. Upon a new (admitted) arrival of thesame class, the capacity is shared among the two. Both areserved in parallel and independently, each with rate µ�/2.Upon a third (admitted) arrival each is served with rate µ�/3.This continues up to K �-agents. Any further arrival, leaveswithout service even after being admitted. When any of theexisting �-agents depart, the service rate is readjusted to anappropriate higher value. The τ -service is resumed only afterall the �-agents depart. We call this as βPSp,K scheduling policy.

Tolerant agents are served in FCFS (first come first serve)basis. They are served in a serial fashion and with fullcapacity3 µτ . That is, system would serve at maximum oneτ -agent, and the service of the next τ -agent begins only afterthe preceding one departs.

The transitions and evolution of the �-agents is independentof that of τ -agents under a static policy: the arrivals areadmitted and the service is provided to the admitted agentsimmediately, irrespective of the state of τ -agents. Thus onecan analyze the �-class independently and we first considerthis analysis.

A. Blocking Probability of �-class

Fix 0 ≤ p ≤ 1, K and consider policy βPSp,K . Blockingprobability is the probability with which a new (�-class) arrivalleaves the system without service. Blocking can occur in caseof two events. Upon arrival, an �-agent is admitted to thesystem with probability p and is blocked with probability(1 − p). Secondly, an admitted agent leaves without service,if the system is already serving K �-agents.

Let Φ�(t) represent the number of �-agents in the systemat time t. We claim that the �-class transitions are causedby exponential random events and hence that Φ�(t) is acontinuous time Markov jump process (see for example [8])for the following reasons: a) it is clear that the inter-arrivaltimes are exponentially distributed with parameter λ�p; b) byLemma 1, given below, the departure times are exponentiallydistributed with parameter µ� (i.e., ∼ exp(µ�)), irrespective ofstate Φ�(t).

Lemma 1: Let Dl� represent the time to first departureamong the l �-agents receiving the service, with 1 ≤ l ≤ K.Then for PS policy, Dl� ∼ exp(µ�) for any l.Proof: When l agents are receiving service in parallel, becauseof processor sharing the service time of each is exponentiallydistributed with parameter µ�/l. And the time to first depar-ture, the minimum of these l exponential random variables, isagain exponential with parameter lµ�/l = µ�. �

In Figure 1, we depict the transitions of the continuous timeMarkov jump process Φ�(t). For such processes, well known

3Capacity of the server is such that, it can either serve one tolerant agentat rate µτ , or l �-agents each at µ�/l (where l ≤ K).

balance equations are solved to obtain the stationary prob-abilities (see for example [8]). The stationary probabilities,{π0, π1, · · · , πK}, of Φ�(t) are obtained by solving:π0λ�p = µ�π1, πl(λ�p+ µ�) = λ�pπl−1 + µ�πl+1 for 1 ≤ l < K,and πKµ� = λ�pπK−1.

The solution or the stationary probabilities are (0 ≤ l ≤ K):

πl =ρl�,pa0

with a0 :=K∑j=0

ρj�,p and ρ�,p :=λ�p

µ�= ρ�p.

An admitted agent gets blocked, if it finds K �-agents inthe system, and, this by PASTA (Poisson Arrivals See TimeAverages) equals the stationary probability πK of K �-agentsin the system. The agents are not admitted with probability(1 − p) and those admitted are blocked with probability πK .Therefore the overall blocking probability equals:

PPSB (p) = (1− p) + pπK = (1− p) + pρK�,pa0

. (7)

B. Expected sojourn time of τ -class

The �-class requires short but frequent jobs (e.g., voicecalls). Hence we are looking for a good relevant approximationthat facilitates the analysis, and which further allows us tostudy other important variants (like CD policy of sectionV). Towards this, we approximately (accurate asymptotically)decouple the evolution of τ -agents from that of �-agents.

We first understand the effective server time (EST), Υτ ,which is defined as the total time period between the servicestart and the service end of a typical τ -agent. We refer this asEST of the agent under consideration, as no other τ -agent hasaccess to server during this period. Sojourn time of a typicalτ -agent equals the sum of two terms: a) waiting time, thetime before the service start; and b) EST Υτ , the time afterthe service start.

1) Analysis of effective server time (EST) (Υτ ): This timeequals the sum of the actual service time, Bτ , of the τ -agentand the overall time of interruptions caused by �-agents, whichis denoted by Υeτ . Let N(Bτ ) represent the total number of the�-class interruptions, that occurred during the service time Bτ .In reality these interruptions would have occurred in disjointtime intervals, the sum of all of which is Bτ . This randomnumber has same stochastic nature as the number of Poissonarrivals that would have occurred in a continuous time intervalof length Bτ . This is true because of the memory less propertyof the exponential service time Bτ and because Poissonprocess is a counting process. After an �-agent interrupts theongoing τ -agent, there is a possibility of further admissions.Eventually the service of the τ -class is resumed, where left,when all the �-agents (that were admitted) leave the system.

Thus the time duration for which the service of τ -agent issuspended per interruption, equals a busy period of the �-class,that started with one �-agent. There would be N(Bτ ) (random)number of such interruptions. Hence,

Υτ = Bτ + Υeτ with Υ

eτ :=

N(Bτ )∑i=1

Ψ�,i , (8)

where {Ψ�,i}i are the IID (independent and identically dis-tributed) copies of �-busy period. We have the following result.

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ML System ML System

O System O System

MU System MU System

Case(a): є- customer present, at τ-arrival Case(b): No є- customer present at τ-arrival

Arrival of τ-customer

and end of its idle

period.

Arrival of є- customer

and beginning of its

busy period.

Service starts for τ-customer

Fig. 2. Example sample paths of the three systems (PS policy)

Average Sojourn time for =-Class Customers0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Bloc

king p

roba

bility

for

0-Clas

s Cus

tomer

s

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Simul. result with 70 = 20.

Simul. result with 70 = 100.

Theoritical result.

Configuration: K=8, 6

==4, 7

= = 8.

;0 = 0.3

;0 = 0.5

Fig. 3. Achievable region: Simulated and Theoretical results

Lemma 2: The first two moments of the �-busy period andEST Υτ are given by:

E[Ψ�] =a1µ�

and E[Ψ2� ] =1

µ2�

K∑i=1

qi−1 ci(1− q)i , (9)

E[Υτ ] =1

µτ+λ�p

µτE[Ψ�] =

a0µτ,

E[Υ2τ ] =2a20µ2τ

+ρ�,pµτµ�

K∑i=1

qi−1 ci(1− q)i ,

where the constants q, {ci} and {ai} are defined as:

ρ�,p =λ�p

µ�, q =

ρ�,pρ�,p + 1

, ai =

K−i∑j=0

ρj�,p for all 0 ≤ i ≤ K, (10)

bi =

K−1∑j=K−i+1

(K − j)ρj�,p for all 2 ≤ i ≤ K, b1 = 0,

c1 =2ρ�,p (2a2 + b2) + 2

(1 + ρ�,p)2µ2�, and for all 1 ≤ i < K

ci =2ρ�,p ((i+ 1)ai+1 + bi+1) + 2 ((i− 1)ai−1 + bi−1) + 2

(1 + ρ�,p)2µ2�,

cK =2ρ�,p(KaK + bK) + 2((K − 1)aK−1 + bK−1) + 2

(1 + ρ�,p)2µ2�.

Proof: The proof is provided in Appendix B. �

2) Approximate decoupling via Domination: Every τ -agentundergoes similar stochastic behaviour, as below. Each τ -agentwill have to wait for the beginning of its service, and has tofinish its service in the midst of random interruptions, all ofwhich have identical stochastic nature. Further, evolution ofthe �-agents during the EST Υτ of one τ -agent is independentof that of the other τ -agents. Hence the Υτ times correspond-ing to different τ -agents are independent of each other. Thusthe idea is to model the τ -class evolution approximately asan independent process, with that of an M/G/1 queue. Thearrivals remain the same, but the service times in M/G/1queue are replaced by the sequence of ESTs {Υtτ}.

We call this M/G/1 queue asML system and the originalsystem as O system. In fact we will define another M/G/1system MU as below and show that: a) the performance(expected sojourn times) of the original system is boundedbetween the performances of the two M/G/1 systems; andb) that the performances of the two sandwiching systemsconverge towards each other as µ� →∞ (even with ρ� fixed).

a) ML system: The ESTs are considered as servicetimes of τ -agent in ML system. We study the (sample pathwise) time evolution of the two systems, original and ML, todemonstrate the required domination. Towards this, we assumethat both the systems are driven by same input (arrival timesand service requirements) processes. Consider that both thesystems start with same number (greater than 0) of τ -agentsand assume that both of them start with service of the firstamong the waiting ones. Then the trajectories of both thesystems evolve in exactly the same manner, until the τ -queuegets empty. There can be a change in the trajectories of the twosystems, upon a subsequent new τ -arrival. We can have twoscenarios as in Figure 2. If �-agents are absent at the τ -arrivalinstance in the original O system (as in sub-figure b), thenagain, both the systems continue to evolve in the same manner.On the other hand, if an �-agent is deriving service (as in sub-Figure a), the service of τ agent is delayed in the original Osystem till the end of the ongoing �-busy period. While theservice starts immediately inML system. Then the trajectoriesin the two systems continue with the same difference, until theend of the next τ -idle period. At this point the difference: a)either gets reduced, if the τ arrival marking the end of τ -idleperiod occurs after sufficient time and finds no �-agent; b) orcan increase, if the τ -arrival occurs again during an �-busyperiod; c) or can continue with almost previous value, if theτ -arrival occurs immediately and finds no �-agent. And thiscontinues. Thus the sojourn times in ML system are lowerthan or equal to that in O system in all sample paths. Aswe notice the difference between the two systems is becauseof �-busy cycles and this difference may diminish if the latershorten. We will show this indeed is true in coming sections.

b) MU system: Consider another M/G/1 system whoseservice times equal Υτ +Ψ�, where Ψ� is an additional �-busyperiod independent of Υτ . It is clear that this system dominatesthe O system everywhere (see O and MU trajectories inFigure 2). Hence the sojourn times of τ -agent in O system areupper bounded by that in MU system (in all sample paths).Thus the expected sojourn time of O system is sandwiched asbelow:

EML [Sτ ] ≤ EO[Sτ ] ≤ EMU [Sτ ]. (11)

3) Performance of ML and MU systems: In Lemma 2,we obtained the first two moments of the �-busy period andthe EST, Υτ . Using the well known formula for the expectedsojourn time of an M/G/1 queue, we have:

8

EML [Sτ ] = E[Υτ ] +λτE[Υ

2τ ]

2(1− ρMLτ )with ρMLτ = λτE[Υτ ].

Similarly with ρMUτ = λτE[Υτ + Ψ�],

EMU [Sτ ] = E[Υτ + Ψ�

]+λτ(E[Υ2τ ] + E[Ψ

2� ] + 2E[Ψ�]EΥτ ]

)2(1− ρMUτ )

.

From Lemma 2 constants {ci}, moments of busy periodE[Ψ�], E[Ψ2� ] converge to zero as µ� → ∞, and so thedifference EMU [Sτ ] − EML [Sτ ] converges to zero. In factthis is true even when µ�, λ� jointly converge to ∞ whilemaintaining ρ� = λ�/µ� constant. If µ� → ∞ for a fixed λ�,then the load factor also decreases to zero in limit. Thus theresult would have been true only for low load factors. But bymaintaining the ratio ρ� fixed when µ� →∞, we ensured thatthe approximation is good for any given load factor and forany given admission control p, i.e., for any (ρ�, p). Under SFJlimit, using Lemma 2:

EPS [Sτ (p)] := EOPS [Sτ (p)] ≈

1

µ̃τ,p(1− ρ̃τ,p), (12)

with ρ̃τ,p = ρτa0, µ̃τ,p =µτa0

and ρτ :=λτµτ.

Thus the achievable region under SFJ limit is given by:

APS ={(

(1− p) + p(ρ�,p)K

a0,

a0µτ (1− a0ρτ )

)∣∣∣ with a0ρτ < 1, 0 ≤ p ≤ 1}.

In the above, condition a0ρτ < 1 ensures stability.

C. Validation of Pseudo conservation law (2), Completeness

By direct substitution4 one can verify that the performancemeasures of βPSp,K scheduler, for every (p,K), satisfy thepseudo conservation law (2). Further as K increases to∞, theblocking probability PPSB (1), given by equation (7), decreasesto zero if ρ� ≤ 1. When ρ� > 1, using simple computations5,one can show that PPSB (1)→ 1−1/ρ� and only pB > 1−1/ρ�can be a part of the Astatichetero. Also it is easy to verify thatthe function, p 7→ PPSB (p), is continuous in p for any K.Thus by intermediate value theorem, all the points of Aheterostaticcan be achieved by these schedulers. And hence the family ofschedulers,

FPS :={βPSp,K , 0 ≤ p ≤ 1,K

},

is complete. It is important to note here that these schedulersachieve the entire static region, nevertheless a larger K impliesa larger time spent by �-agents in the system. Thus system mayhave a restriction on the size of K to be used based on theQoS requirements.

4By (7), 1−ρ�(1−PPSB ) = 1/a0 and so(µτ[1−ρ�(1−PPSB )

]−λτ

)−1(see equation (2)) equals EPS [Sτ ] given by (12).

5It is easy to verify as K →∞ that:ρK�∑Kl=0 ρ

l�

=1∑K

l=0 ρ−(K−l)�

=1∑K

l=0 ρ−l�

→11

1−ρ−1�

= 1−1

ρ�.

p p p p p . . .

K

0 1 2 K-1 K

Fig. 4. State transitions for �-agents in CD model.

V. CAPACITY DIVISION (CD) POLICIES

In the previous section, when an admitted �-customer pre-empts the ongoing service of τ -customer the entire systemcapacity is transferred to �-customer. In this section we analyzea different scheduling policy. Here the capacity is not com-pletely transferred, but rather a fraction of it is used by each�-customer. The τ -customer is continued with the remainingcapacity.

Service Discipline: Each �-customer uses (1/K)-th part ofthe capacity, µ�/K. If the system has only one �-customer, theremaining capacity i.e., (K − 1)/K-th part of the capacity isutilized by the τ -customer. In other words, τ -class is servedwith rate µτ (K − 1)/K. If there are 0 ≤ l ≤ K numberof �-customers receiving the service, then (l/K)-th part ofthe capacity is used by the �-customers and the τ -customeris served at rate ((K − l)/K)µτ . This continues up to K �-customers, and any further (admitted) �-arrival, departs withoutservice. Whenever an existing �-customer departs, the capacityis readjusted to an appropriate higher value for τ -customer.

It is clear that �-class evolution is again independent of τ -class evolution and its analysis is considered first.

A. Blocking Probability of �-class

Consider any fixed 0 ≤ p ≤ 1. The �-inter arrival times areexponentially distributed with parameter λ�p. Say there are l�-agents in the system (note l ≤ K). Each one of them receiveservice at rate µ� and this happens simultaneously. Thus thefirst departure time would be exponentially distributed withparameter lµ�. This is again a continuous time Markov jumpprocess and its transitions are as shown in Figure 4. In fact the�-agents evolve like the well known, finite capacity and finitebuffer queueing system, M/M/K/K queue. The stationarydistribution of such a queue is well known and in particular(see for e.g., [8]):

π̌K =(Kρ�,p)

K

K!ǎ0where ǎ0 :=

K∑j=0

(Kρ�,p

)jj!

.

As before, agents are admitted with probability (1 − p), andhence the overall blocking probability by PASTA equals:

PCDB (p) = (1− p) + pπ̌K = (1− p) + p(Kρ�,p)

K

K!ǎ0.(13)

B. Expected sojourn time of τ -class

The idea is once again to approximately decouple theevolution of τ -agents from that of �-agents. The procedureis similar, however the current model is more complicated.Once again EST is denoted as Υ̌τ , has similar meaning as insection (IV-B) and typical τ -sojourn time equals the sum ofwaiting time and the effective server time (EST).

9

ϒ� � starts

ϒ� � departs

ϒ� �U � ϒ� �U ��

Upper Dominating System

ϒ� � starts

ϒ� � departs

ϒ� �L � ϒ� �L ��

Lower Dominating System

Fig. 5. Lower and Upper dominating systems (CD Model)

1) Analysis of effective server time ˇ(Υτ ): In the CD model,EST is the total time period between the service start andservice end of a typical τ -agent, when the capacity is dividedpossibly between the two classes. With the arrival+admissionof each �-agent the server capacity available for the ongoingτ -agent reduces, with each �-departure it increases, and itsservice is completed in the midst of such rate changes. Infact, the τ -agent’s service is completely pre-empted withthe admission of K-th �-agent. The service would again beresumed, where left, when one of the K �-agents depart. TheEST depends upon the number of �-agents in the system at theservice start. Hence we introduce superscript l in the notationof Υ̌. That is, Υ̌lτ represent the EST, when it starts with l �-agents. Thus the analysis of EST for this model is not as easyas in PS model. One can not estimate this using the numberof interruptions and time per interruption as in PS model.However, the underlying transitions are Markovian in nature,and hence we obtain the analysis by directly considering theEST’s {Υ̌lτ}l. We have the following, (proof in Appendix C):

Theorem 2: The first two moments of EST Υ̌0τ are:

E[Υ̌0τ ] =ǎ0 +O(1/µ�)

ηµτ +O(1/µ�), ǎ0 :=

K∑j=0

(ρ�,p)j

j!and (14)

E[(

Υ̌0τ)2]

=2ǎ20ηµτ

+O(1/µ�)

ηµτ +O(1/µ�)with η :=

K−1∑j=0

(ρ�,p)j

j!

K − jK

,

where f(µ�) = O(1/µ�) for any function f implies,f(µ�)µ� → constant as µ� →∞, with ρ� fixed. �

2) Dominating systems: It was not difficult to obtain theconditional moments of the EST, {Υ̌lτ}l, when conditionedon the number of �-agents at the service start. However toobtain the unconditional moments, one requires the stationarydistribution of the number at service start of a typical τ -agent. And this is not a very easy task. However the variousconditional moments differ from each other at maximum inone �-busy period. Hence one can possibly obtain the (ap-proximate) unconditional moments, along with M/G/1 queueapproximation, using the idea of dominating fictitious queues.

We again have two dominating systems, which dominateon either side at the beginning of the τ -busy period, exactlyas in the previous model. In addition, two partial �-busyperiods (if possible) are subtracted from every EST in thelower dominating system. While the upper dominating systemis obtained by adding two extra partial �-busy periods as inFigure 5. Using exactly the same logic as in the previousmodel, one can show that the expected sojourn time of the CDmodel can also be obtained as limit of the expected sojourn

times of M/G/1 queues with service time moments given bythat of Υ̌0τ of Theorem 2. We again consider the SFT limit.

3) Sojourn time of CD Model: Using M/G/1 analysis andTheorem 2, the sojourn time for τ -agent in SFJ limit:

ACD =

{((1− p) + p (Kρ�,p)

K

K!ǎ0,

1

µ̈τ,p (1− ρ̈τ,p)

)(15)

: ρ̈τ,p < 1, 0 ≤ p ≤ 1

}, with ρ̈τ,p = λτµ̈τ,p ,

ǎ0 :=

K∑j=0

(Kρ�,p

)jj!

, η :=

K−1∑j=0

(Kρ�,p

)jj!

K − jK

, and µ̈τ,p =ηµτǎ0

.

By direct substitution6 one can verify that the CD policiesalso satisfy the pseudo conservation law (2). Further they alsoform a complete family of schedulers, for exactly the samereasons as that for PS policy when ρ� ≤ 1 and using Lemma5 of Appendix E.

VI. NUMERICAL EXAMPLES

Random system with large µ�: We conduct Monte-Carlosimulations to estimate the performance of both the policies.We basically generate random trajectories of the two arrivalprocesses, job requirements and study the system evolutionwhen it schedules agents according to PS/CD policy. Weestimated the blocking probability and expected sojourn timefor � and τ -agents respectively, using sample means, fordifferent values of (p,K).

In Figure 3, we consider an example to compare thetheoretical expressions with the ones estimated using Monte-Carlo simulations for PS policy. We consider two differentvalues of ρ�. We notice negligible difference between thetheoretical and simulated values with µ� = 100. However evenwith µ� = 20, the difference is about 10-12% for most of thecases.

We consider another example of PS model in Table I with,K = 8, λτ = 4, µτ = 8 and ρ� = 0.5. As the service rateof �-agents increases with fixed load factor (ρ�), the simulatorresults are close to the theoretical results. The performanceis very close to that of theoretical, for values of µ� greaterthan 120. For µ� = 80, the difference is at maximum 5%.Even at values as low as 20, the simulator performance iswithin 10% of the theoretical values for most of the cases.Thus the theoretical results well approximate the simulatedones, in most of the scenarios. Especially in the cases withlarge µ�, λ�.

Achievable region: is also plotted in Figure 3 for differ-ent values of ρ�. Towards this, we plot EPS [Sτ (p)] versusPPSB (p), for p ∈ {iδ : 0 ≤ i ≤ 1/δ} with sufficiently smallδ > 0. It is a convex curve. We notice a downward shift(improvement) in the curve with smaller ρ�, as anticipated.However the formula derived, helps us understand the exact

6From equation (13),

1− ρ�(1− PCDB ) =∑K−1j=0

(Kρ�,p

)jj!

K−jK

K∑j=0

(Kρ�,p

)jj!

=η

ǎ0

and so(µτ[1−ρ�(1−PPSB )

]−λτ

)−1 (see equation (2)) equals EPS [Sτ ]given by (15).

10

Average Sojourn time 10 -1 10 0 10 1 10 2

Bloc

king

prob

abilit

y

10 -4

10 -3

10 -2

10 -1

10 0

PS ModelCD Model

K=3

K=5

Fig. 6. Achievable regions ACD , APS : PB versus E[Sτ ] for different ρ�, ρ� = 0.9/K

0 0.2 0.4 0.6 0.8 1Probability of admission p

10-2

10-1

100

Blocki

ng Pr

obab

ility

PS ModelCD Model

0 0.2 0.4 0.6 0.8 1Probability of admission p

0

5

10

15

20

25

Expe

cted S

ojourn

Time

PS ModelCD Model

Fig. 7. PB , E[Sτ ] versus p

Simulation Theoreticalp µ� PPSB E[Sτ ] P

PSB E[Sτ ]

020 1.0000 0.2500 1.0000 0.250080 1.0000 0.2500 1.0000 0.2500120 1.0000 0.2500 1.0000 0.2500

0.2520 0.7500 0.3525 0.7500 0.333380 0.7500 0.3387 0.7500 0.3333120 0.7500 0.3373 0.7500 0.3333

0.5020 0.5000 0.5671 0.5000 0.500080 0.5000 0.5146 0.5000 0.5000120 0.5000 0.5105 0.5000 0.5000

0.7520 0.2502 1.2377 0.2502 0.999380 0.2502 1.0417 0.2502 0.9993120 0.2502 1.0367 0.2502 0.9993

120 0.0020 110.2218 0.0020 127.75080 0.0020 112.1243 0.0020 127.750120 0.0020 127.1987 0.0020 127.750

TABLE IPS MODEL: COMPARISON OF SIMULATED, THEORETICAL METRICS

amount of shift. We plotted the curves only in the τ -stabilityregion, {λτ : a0ρτ < 1}. Unlike the case of homogeneousagents, the τ -stability region varies with the scheduling policy.This is because, varying fractions of �-agents are lost withdifferent values of p, which can expand or contract the stabilityregion.

Comparison of the two policies: We compare the achievableregions of PS and CD policies by plotting ACD and APS .We set ρ� = 0.9/K, λτ = 5.6 µτ = 8 and K = 3 or 5.

In Figure 6, we plot the achievable region for both themodels/policies, i.e, we plot E[Sτ (p)] versus PB(p), for dif-ferent p. And in Figures 7, we plot the performance measuresPB(p) and E[Sτ (p)] respectively versus p with K = 3. FromFigure 6, the two achievable (sub) regions overlap, howeverwe observe from the Figures 7 that the performance measuresof the two models are different for the same (p,K). Butif we choose a p and p′ such that PCDB (p) = PPSB (p′),we observe that the two expected sojourn times are equal.Because of this the two achievable regions overlap in Figure6. This observation is precisely the pseudo-conservation law.Whatever the policy used, once the blocking probabilities arethe same the expected sojourn times are the same.

Now we will discuss a slightly different, yet, a relatedimportant aspect. We would compare the two sets of policies,when K (maximum number of parallel calls) is the same. Asseen from the figures the sub-achievable region of CD policy,with fixed K, is a strict subset of that of the PS policy. This is

because the best possible blocking probability with CD policy,

PCDB (1) =(Kρ�)

K/K!∑Kj=0 (Kρ�)

j/j!≥ (ρ�)

K∑Kj=0(ρ�)

j= PPSB (1),

is greater than that with the PS policy. In Figure 6 the bestPB with CD and PS models/policies respectively is 0.002and 0.0002 (0.05 and 0.019) when K = 5 (K = 3). Thusit appears that the static achievable region would overlap fordifferent policies, however the sub-regions covered by differentpolicies can be different when K is fixed.

0.17 0.19 0.21 0.23 0.2510-4

10-3

10-1

100

Blo

ckin

g pr

obab

ility

PS PolicyCD PolicyPseudo conservation law (2)

Expected Sojourn Time

;0 = 0.2 6

= =2.4

7= = 8 K=3

Fig. 8. Static Achievable region

Expected Sojourn time 0.17 0.19 0.21 0.23 0.25

Blo

ckin

g p

robabili

ty

10 -4

10 -2

10 0Configuration: K = 50 ;

= = 0.3 ;

0 = 0.2

-p,KPS policy

Pseudo Conservation law (2)

Fig. 9. Completeness of FPS

Completeness: In Figure 8, we plot pseudo-conservationlaw (2). We also plot the performance of PS/CD policies withK = 3 and for varying p. We see that the three curves exactlyoverlap, again validating (2). For the same configuration weplot performance of PS policies with a bigger K = 50, inFigure 9. With K = 50 we are able to achieve a bigger partof the achievable region. One can achieve a similar result withCD policy. With even bigger K one can achieve further lowerparts of the pseudo-conservation curve. However, as mentionedbefore, one may not be able to use a larger K because of otherQoS restrictions. For example, the � customers may not agreefor a very small service rate (µ�/K) which can prolong theirstay in the system. It is in this context that the PS couldbe better than the CD policies. Even though both the sets ofpolicies are complete, PS policy achieves a bigger sub-regionthan the CD policy for the same K (see Figures 6 and 8).

VII. A DYNAMIC POLICY

We consider dynamic policies (for PS model) with an aimto demonstrate that the dynamic region is bigger than the staticregion. Towards this we construct an example dynamic policy

11

and show that the block probability, for the same sojourn timeE[Sτ ], is better with the dynamic policy.

The static policy of the previous sections is modified asfollows. We refer this as policy βdp . When there are no τ -agents in the system, i.e., during the τ -idle period, there is noadmission control for �-agents. An arriving �-agent is admittedwith probability one. Recall, however that service is offeredto an admitted agent only when the number in system is lessthan K. When the system is in τ -busy period7, i.e., when theτ -queue is non-empty, we admit the �-agents with probabilityp. So, this is a dynamic policy which alternates between fulland partial admission.

Let Ψτ and Iτ respectively represent the busy and idle peri-ods of the τ -agents. By stationarity, memoryless property, theconsecutive busy, idle periods {Ψτ,i}, {Iτ,i}i are independentand identically distributed. We have (proof is in Appendix D):

Theorem 3: The block probability, PBd (p), for the systemwith the dynamic policy βdp :

PBd (p) =E[Iτ,1]PB(1)

E[Ψτ,1] + E[Iτ,1]+

E[Ψτ,1]PB(p)

E[Ψτ,1] + E[Iτ,1]. � (16)

Using the ideas of dominating systems as in the section IVone can show that the moments of the idle, busy periods ofthe original system with policy βdp converges towards that ofthe equivalent M/G/1 system ML, as µ� → ∞. Thus wewill have for large values of µ�:

E[Iτ,1] ≈1

λτ,

EO[Ψτ,1] ≈ EML [Ψτ,1] =E[Υτ ]

1− λτE[Υτ ]→ a0

µτ − λτa0.

The second last equality is obtained using the well knownformula for the average busy period of an M/G/1 queue. Itis easy to see that the sojourn time of the dynamic policy βdp issame as that with static policy βp (asymptotically), while theblocking probability is improved from (7) to (16). Note thatPB(1) ≤ PB(p) for any p ≤ 1. Hence the dynamic policyperforms better and the dynamic achievable region is bigger.One can obtain similar improvement with CD model.

Numerical comparison of Dynamic and Static regions

In Figure 10, we compare the performance of the dynamicpolicy βdp with the corresponding static policy, for PS model.We notice a good improvement in the curve: blocking prob-ability decreases significantly for the same expected sojourntime. This indicates that the dynamic region is strictly biggerthan the static region, unlike the homogeneous case. In fu-ture, we would like to obtain complete analysis of dynamicachievable region for this heterogeneous system.

VIII. CONCLUSIONS AND FUTURE WORK

We consider a queueing system with heterogeneous classesof agents. The impatient class demands immediate service,hence receives the service immediately and if required in

7Normally a busy period begins immediately with an arrival to an emptyqueue. However, in our system we say a τ -busy period starts with the servicestart of that τ -agent, which arrives to an τ -empty queue. If �-agents werepresent at the τ -arrival instance, the service of the τ -agent is deferred till theend of the ongoing �-busy period.

10−1 100 101 102

.4

.6

.8

1

Average Sojourn time for τ−Class Customers.

Bloc

king p

roba

bility

for ε−C

lass C

ustom

ers.

λτ = 5, static policy

λτ = 5, dynamic policy

λτ = 4, static policy

λτ = 4, dynamic policy

λτ = 5

λτ = 4

Configuration: µτ=7,λ

ε=2,µ

ε=3.

Fig. 10. Static-dynamic policies, K = 4.

parallel with others. There is an admission control to ensurethe QoS requirements of the other (tolerant) class. The tolerantclass can wait for their turn, however would like to optimizetheir sojourn time.

We conjecture a pseudo conservation law for this lossyqueueing system, which relates the blocking probability ofimpatient agents to the expected sojourn time of the tolerantagents, in a short and frequent job (SFJ) limit-regime for theformer. The pseudo conservation law should be satisfied by allthe policies, that are static (do not depend upon the state) andwork conserving (left over server capacity is completely usedwhen there is a customer) with respect to the tolerant agents.

We consider two families of scheduling policies, whichdiffer in the way the system capacity is shared between thetwo classes. With processor sharing policy the entire systemcapacity is transferred to impatient customer, once admitted.In the second policy, which we refer as capacity divisionpolicy, only a (fixed) fraction of capacity is transferred to eachadmitted impatient customer.

We obtain closed form expressions for the asymptoticperformance measures, under SFJ limit, for both the familiesof policies. The two families satisfy the pseudo-conservationlaw. Further, both the families are complete, i.e., they attainevery point of the achievable region given by the pseudo-conservation law. The CD achievable region is a strict subsetof the PS region, when restricted to the same number ofparallel service possibilities. This demonstrates the limitationof CD model, which could be a more practically used model.The PS model can attain a much smaller blocking probability.

We obtain the performance with an example dynamic policyand illustrate that the dynamic region is strictly bigger than thestatic region. This is in contrast to the homogeneous case (alltolerant classes), where the two regions coincide. However,under a dynamic policy the impatient agents may experiencenon-stationary blocking. Some systems may not prefer this andhence static policies have importance of their own.

12

The results are asymptotic and are accurate when the arrival-departure rates of the impatient class is large. Usually suchcustomers have short frequent job requirements and hence thisis an useful asymptotic result. Further, we have an upper andlower bound for the sojourn time performance, even when therates are not large.

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[12] J. G. Shanthikumar and D. D. Yao, ‘Multiclass queueing systems:Polymatroidal structure and optimal scheduling control’, OperationsResearch, vol. 40, no. 3-supplement-2, pp. S293–S299, 1992.

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[14] Sleptchenko, A., A. van Harten, and M. C. van der Heijden. ‘Anexact solution for the state probabilities of the multi-class, multi-serverqueue with preemptive priorities’, Queueing Systems, 50.1, 2005.

[15] A. Federgruen and H. Groenevelt, ‘M/G/c queueing systems withmultiple agent classes: Characterization and control of achievable per-formance under nonpre-emptive priority rules’, Management Science,vol. 9, pp. 1121– 1138, 1988.

[16] D. Bertsimas, I. Paschalidis, and J. N. Tistsiklis, ‘Optimization ofmulticlass queueing networks: Polyhedral and nonlinear characteriza-tions of achievable performance’, The Annals of Applied Probability,vol. 4, pp. 43–75, 1994.

[17] D. Bertsimas, ‘The achievable region method in the optimal controlof queueing systems; formulations, bounds and policies’, Queueingsystems, vol. 21, no. 3-4, pp. 337–389, 1995.

[18] D. Bertsimas and J. Niño-Mora, ‘Conservation laws, extended poly-matroids and multiarmed bandit problems; a polyhedral approach toindexable systems’, Mathematics of Operations Research, vol. 21,no. 2, pp. 257–306, 1996.

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PROOFS AND APPENDICES ARE IN THE NEXT PAGE

13

APPENDIX : STATIC AND DYNAMIC REGIONS COINCIDE IN HOMOGENEOUS SYSTEMS

We discuss the case without pre-emption. By the well known work conservation principle ([9]), for any (work conserving)scheduling policy the expected waiting times of the two classes satisfy:

ρ1E[W1] + ρ2E[W2] = c =⇒ E[W1] =c

ρ1− ρ2ρ1E[W2], (17)

where c is an appropriate constant. A family of schedulers is called complete (e.g., [20], [21]), if every performance vector ofthe achievable region is obtained by one of its schedulers. It is well known that the priority schedulers (see for e.g., [3]) forma complete family for homogeneous classes. Each scheduler is represented by a priority factor b, say meant for class 1, andclass 1 is scheduled if b times its longest waiting time w̃1 is greater than w̃2. By varying b from 0 to infinity, we obtain allthe performance pairs of the dynamic region.

Consider the set of static policies parametrized8 by p ∈ (0, 1). Class 1 is scheduled with probability p, now independent ofeverything else. As p→ 1, class 1 gets maximum priority. It is not difficult to see that the limit performance of the two classeswith p → 1 equals that obtained with dynamic priority scheduler as b → ∞. Both the limits are obtained by giving absolutepriority to class 1 and hence are equal. Similarly the performance under static policy with p→ 0, equals the performance underdynamic policy with b = 0. Further, it is easy to show that the performance of the two queues is continuous in parameter p ofthe static policy. Thus by intermediate value theorem all the values between the two extremes are achieved as p varies between0 and 1. And these pairs satisfy (1). Hence the static achievable region coincides with dynamic region in homogeneous setting.�

APPENDIX A: RESULTS RELATED TO PSEUDO CONSERVATION LAW

Lemma 3: Let τµ := inft{t : Rµ(t) ≥ B

}, be any random time defined using any random variable B which is independent

of {A1�,n, B1�,n}n. Since Rµ(t) satisfies (6) we have:∣∣∣τµ − Bντ

∣∣∣→ 0 a.s. as µ→∞Proof: By continuity of probability measure, one can assume that (6) is satisfied together for a sequence of {Wn} withWn →∞, almost surely. Let

A :=

{w : sup

t∈[0,Wn]

∣∣Rµ(t)− ντ t∣∣→ 0 for all n} and P (A) = 1.For any w ∈ A, consider a Wn > (B(w) + 1)/ντ . For every 1 > ε > 0, there exists an µε

14

Proof: For any � there exists a T� such that: ∣∣∣∣R1(t)t − a∣∣∣∣ ≤ � ∀ t ≥ T� (18)

Consider s such that µs ≥ T�, it follows that ∣∣∣∣R1(µs)µs sW − a sW∣∣∣∣ ≤ sW � ≤ �. (19)

If s is such that µs < T�, ∣∣∣∣R1(µs)µs sW − a sW∣∣∣∣ = ∣∣∣∣R1(µs)µW − a sW

∣∣∣∣≤ R

1(µs)

µW+ a

s

W

≤ R1(T�)

µW+ a

T�µW

< �, (20)

by choosing µ further large to get the required upper bound, if required. Equations (19) and (20) are true for any arbitrary s.Hence the result follows. �

APPENDIX B: PROOF OF LEMMA 2

By conditioning on Bτ , one can verify that

E[N(Bτ )] =λ�p

µτ, E[BτN(Bτ )] =

2λ�p

µ2τ,

E[(N(Bτ ))2] =

λ�p

µτ+

2(λ�p)2

µ2τ.

By conditioning on N(Bτ ) we obtain the first moment:

E[Υτ ] = E[Bτ ] + E

N(Bτ )∑i=1

Ψ�,i

(21)= E[Bτ ] + E

EN(Bτ )∑

i=1

Ψ�,i

∣∣∣∣∣∣N(Bτ ) = 1

µτ+λ�pE[Ψ�]

µτ.

Note that the busy periods {Ψ�,i}i are IID. From (8) we have:

E[Υ2τ ] = E[B2τ ] + 2E

[BτΥ

eτ

]+ E

[(Υeτ )

2]. (22)By first conditioning on (Bτ , N(Bτ )) and then on Bτ :

E[BτΥ

eτ

]= E

EBτ N(Bτ )∑

i=1

Ψ�,i

∣∣∣∣∣∣Bτ , N(Bτ )

= λ�pE[Ψ�]E

[BτBτ

]=

2λ�pE[Ψ�]

µ2τ. (23)

Conditioning as before and because of independence:

E[(Υeτ )

2] = EE

(N(Bτ )∑i=1

Ψ�,i

)2∣∣∣∣∣∣N(Bτ ) ,

=λ�pE

[Ψ2�]

µτ+

2(λ�p)2

µ2τ

(E[Ψ�])2

,

which simplifies to (9).

15

Busy period of �-class: Busy period of any class is defined as the time till the first epoch at which all the customers ofthat class have departed. Let Ψk, represent the busy period of �-class, when it begins with k number of customers. Note thatΨ� = Ψ1. In all the discussions below, an arrival is meant an admitted arrival.

The busy period Ψ1 starts with the arrival of one �-customer. If the customer leaves before the next arrival, the busy periodends. On the other hand, if an arrival occurs before the departure of the existing customer, it marks the beginning of a busyperiod with two customers, Ψ2. As seen in section IV-A (see Fig. 1), a departure time is memoryless, i.e., exponential randomvariable with parameter µ� irrespective of the number of customer sharing the service. Let D represent the departure time.The inter arrival time, A, is exponential with parameter λ�p. Let W := min{D,A} represent the minimum of the two. Withthese definitions:

Ψ1 = 1{D

16

APPENDIX C: PROOF OF THEOREM 2Let Iml := [l, · · · ,m] represent the interval of integers, the big O notations are shortly represented by

O� := O(1/µ�), O(2)� := O(1/µ

2�), λ̃ := λ�p and [i] := K − i. (28)

Let Υ̌l = Υ̌lτ , with l ∈ IK−10 , represent a simpler notation for EST when it begins with l �-agents. Let us begin with Łthe

analysis of Υ̌0. If τ -agent leaves before next �-arrival, the EST ends. If instead an �-agent arrives before, it marks the beginningof EST, Υ̌1.Ł Let Dτ represent the departure time of τ -agent and Dτ ∼ exp(µτ ). It equals Bτ since the service is offeredat full capacity. Let A ∼ exp(λ̃) represent the exponential inter arrival time of admitted �-agent. Let W̄0 := min{Dτ , A}represent the minimum. Then,

Υ̌0 = 1{Dτ

17

Simplifying we obtain:

E[(

Υ̌K)2]

= rK + E[(

Υ̌K−1)2] with δK = γK = Kµ� and (36)

rK =2

δKαK+σKδK

, σK =2δKαK

E[Υ̌K−1

]+

2λ̃

αKE[Υ̌K].

Similarly from (30) we obtain for any i ∈ IK−11 ,

E[(

Υ̌[i])2]

= r[i] +[i]µ�δ[i]

E[(

Υ̌[i]−1)2] with (37)

r[i] =1

δ[i]

[2

α[i]+ λ̃r[i]+1 + σ[i]

],

δ[i] =α[i]δ[i]+1 − λ̃([i] + 1)µ�

δ[i]+1,

σ[i] =2λ̃

α[i]E[Υ̌[i]+1] +

2([i])µ�α[i]

E[Υ̌[i]−1].

From (29),E[(

Υ̌0)2]

=λ̃

α0E[(

Υ̌1)2]

+2

α20+

2λ̃E[Υ̌1]

α20.

Further using equation (37) with i = K − 1 or [i] = 1:

E[(

Υ̌0)2]

=1

δ0

[2

α0+ λ̃r1 +

2λ̃E[Υ̌1]

α0

], δ0 =

α0δ1 − λ̃µ�δ1

. (38)

SFT Limit: From (35) and using (41) of Lemma 4 (see (28)):

E[Υ̌0]

=ǎ0 +O�ηµτ +O�

. (39)

Thus and considering the limit (forward) recursively in (32)

E[Υ̌l]

= limE[Υ̌0]

+O� =ǎ0ηµτ

+O�.

Hence for all i ∈ IK−10 from (37)

σ[i] = θ +O�, rK λ̃ =ρ�,pθ

K+O� where θ :=

2ǎ0ηµτ

.

Again considering limits (backward) recursively in (37), while using the above two equations and equation (42) of Lemma 4and backward induction (as in Lemma 4) we obtain:

λ̃r[i] =

(ρ�,p[i]

+O�

)(λ̃r[i]+1 + θ +O�

)=

(ρ�,p[i]

+O�

)(i−1∑j=0

ρj+1�,p θ

([i] + 1) · · · ([i] + 1 + j) + θ +O�

)

=

i∑j=0

ρj+1�,p θ

[i] · · · ([i] + j) +O� for all i ∈ IK−10 .

Simplifying we obtain:λ̃r1 +

2λ̃E[Υ̌1]

α0=

2ǎ20ηµτ

+O�.

Further using δ0 of (42) of Lemma 4 we obtain the asymptotic limit of the second moment (38). �Lemma 4: We have the following asymptotic results for the coefficients defined in the proof of Theorem 2:

n[i] = 1−µτω[i]µ�

+O(2)� , i ∈ IK−11 with (40)

ω[i] :=1

K

i−1∑j=0

(i− j)ρj�,p([i] + j)([i] + j − 1) · · · ([i]) ,

λ̃+ µτ − n1λ̃ = η +O�, 1 +m1λ̃ = ǎ0 +O�, (41)

δ[i] = [i]µ� + [i]ω[i]µτ +O� ∀ i ∈ IK−11 and δ0 = ηµτ +O�. (42)

Proof: We begin with terms {n[i]} and prove the required result by backward mathematical induction. From (34),

nK−1 = 1−µτωK−1µ�

+O(2)� , with ωK−1 :=1

K(K − 1) .

18

Assume the statement holds for i = l − 1, i.e., say:

nK−l+1 = 1−µτωK−l+1

µ�+O(2)� and

ωK−l+1 :=1

K

l−2∑j=0

(l − 1− j)ρj�,p(K − l + 1 + j) · · · (K − l + 1) .

We need to prove the result for i = l. From (34) and substituting the above

nK−l=(K − l)µ�

αK−l − nK−l+1λ̃=

(K − l)µ�(K − l)µ� + lµτK + µτρ�,pωK−l+1 +O�

= 1− µτωK−lµ�

+O(2)� as µ� →∞ with ρ� constant.

This proves (40). It is easy to see that

ω1 =

K−2∑j=0

K − jK

ρj�,pj!

and hence that ω1ρ�,p + 1 = η,

where η is defined in the hypothesis of the Theorem 2. Using this we obtain the first part of (41):

λ̃+ µτ − n1λ̃ = (ω1ρ�,p + 1)µτ +O� = ηµτ +O�.

Using (40), for all i ∈ IK−12 (note λ� = ρ�µ� with ρ� fixed),

α[i] − n[i]+1λ̃ = [i]µ� +(ω[i]+1ρ�,p +

i

K

)µτ +O� (43)

and hence λ̃α[i] − n[i]+1λ̃

=ρ�,p[i]

+O�.

From above and using a similar backward induction on {mi} of (34) we obtain,

m[i]λ̃ =

i+1∑j=1

ρj�,p([i])([i] + 1) · · · ([i] + j) +O�, ∀ i ∈ I

K−12 . (44)

Thus we get the second part of (41):1 +m1λ̃ =

K∑j=0

ρj�,pj!

= ǎ0 +O�.

Let ζ[i] := [i]µ�/δ[i], for all i ∈ IK−11 . Using the recursive definition of δi, as given in (37), ζ[i] satisfies the following recursiveequation,

ζ[i] =[i]µ�

α[i] − λ̃ζ[i]+1, (45)

just like the recursive definition of {ni} given in (34). Further,

ζK−1 = (K − 1)µ�/γK−1 = nK−1 and hence using (40)

ζ[i] = n[i] = 1−µτω[i]µ�

+O(2)� for all i ∈ IK−11 .

Thus we have the first part of (42):

δ[i] = [i]µ� + µτ [i]ω[i] +O� for any i ∈ IK−11 .

And from (38),δ0 = µτ + µτρ�,pω1 +O� = ηµτ +O�. �

APPENDIX D: PROOF OF THEOREM 3

For any static policy βp, the processor sharing system with �-agents is ergodic. With Lp(T ) representing the number of �agents lost in time T , when the arrivals are admitted at rate p, we have:

limT→∞

Lp(T )

T= PB(p) almost surely,

where PB(p) is given by equation (7).Let Ldp(T ) represent the number of �-agents lost in time T with dynamic policy. Let Iτ (T ), Ψτ (T ) respectively represent

the total τ -idle time and total τ -busy period until time T . These are basically the sum of all the busy/idle periods that elapsedtill the time T . Note that Iτ (T ) + Ψτ (T ) = T. In this case,

Ldp(T ) = Ldp(Iτ (T )) + Ldp(Ψτ (T )).

19

At the end epoch of any busy period (Ψτ,i for some i), the system is completely empty. That is, agents of both the classesare absent. Also a τ -busy period starts only once the system is free from all of its �-agents. Thus there are no �-agents in thesystem at both start and end epochs of a τ -busy period. Hence the evolution of the �-class loss counting process that occurredduring disjoint time intervals (of τ -busy periods) constituting Ψτ (T ) is stochastically equivalent to the �-class loss countingprocess that would have evolved in a continuous time interval of length exactly Ψτ (T ). This is because of the memorylessproperty associated with Poisson arrival process. Thus we have:

Ldp(Ψτ (T ))

Ψτ (T )→ PB(p) almost surely .

In a similar way at the start/end epoch of any τ -idle period the system is free of �-agents. Using similar arguments, andbecause all �-agents are admitted during idle periods we have:

Ldp(Ψτ (T ))

Ψτ (T )→ PB(1) almost surely .

Further using renewal reward theorem, one can show that the following happens almost surely:

Ψτ (T ))

T→ E[Ψτ,1]

E[Ψτ,1] + E[Iτ,1],Iτ (T ))T

→ E[Iτ,1]E[Ψτ,1] + E[Iτ,1]

.

Using all the results established so far, equation (16) follows because:

PBd (p) = limT→∞

Ldp(T )

T. �

APPENDIX ELemma 5: For any ρ� > 1,

PCDB (1)→ 1−1

ρ�and PCDB (1)→ 0 if ρ� ≤ 1.

Proof: When ρ� ≤ 1 we have:

PB(1) =(Kρ�)

K

K!K∑j=0

(Kρ�

)jj!

=

1

K∑j=0

(Kρ�

)jj!

(Kρ�)K

K!

=1

K∑j=0

K(K−1)···(K−j+1)Kj ρ

j−K�

≤ 1K∑j=0

K(K−1)···(K−j+1)Kj

.

Let

f(K) :=

K∑j=0

K(K − 1) · · · (K − j + 1)Kj

= 1 + (1− 1K

) + (1− 1K

)(1− 2K

) + · · ·+ ΠKi=1(1−i

K),

and note thatf(K) ≥ NΠNi=1(1−

i

K) for any N ≤ K.

Fix any ε > 0. For any N there exist a large KN such that for all K ≥ KN ,

((1− 1K + 1

) · · · (1− NK + 1

) ≥ (1− ε) which implies f(K) ≥ N(1− ε) and hence f(K)→∞ as K →∞.

Thus PCDB (1)→ 0. When ρ� > 1, it is clear after redefining that:

f(K) :=

K∑j=0

K(K − 1) · · · (K − j + 1)Kj

ρj−K� ≤K∑j=0

ρj−K� =

K∑j=0

ρ−j� hence limK→∞

f(K) ≤ 11− ρ−1�

.

On the other hand for any ε > 0, as before for any N for all K > KN we have:

f(K) ≥K∑

j=K−Nρj−K� (1− ε) =

N∑j=0

ρ−j� (1− ε).

By first letting N →∞ we have

limK→∞

f(K) ≥ (1− ε)(

1

1− ρ−1�

)and then with ε→ 0 lim

K→∞f(K) ≥ 1

1− ρ−1�. �