Stochastic Bounds for Queueing Systems with Multiple Markov Modulated Sources Zhen Liu INRIA Centre Sophia Antipolis 2004 Route des Lucioles B.P. 93 06902 Sophia Antipolis, France Don Towsley Department of Computer Science University of Massachusetts Amherst, MA 01003-4610 We turn our attention now to Poisson sources. In In this paper we consider a single server queue being fed by a population of identical and indepen- this case, the results are somewhat limited. We dent Markov modulated sources. We are interested restrict our attention to ON-OFF sources where in comparing the behavior of this system when the the O F F state holding times are exponentially disnumber of sources is N to that of the system when tributed and the ON state holding times are charthe number of sources is N + i and the rate at which acterized by a distribution with decreasing failure work arrives to the system has been scaled to match rate (DFR). Customers arrive according to a Poisthat of the first system. We consider two variations son process with rate A while in the ON state. that differ according to whether the sources are fluid Under these assumptions, we again consider two systems, one with N such sources, the second with sources or Poisson sources. We begin by summarizing our results for fluid N + 1 in which the arrival rate for a source while sources. Each source can be in any one of a count- in the ON state is N/(N + 1)A. In both cases, able number of states. The state holding times form we assunm that service times are exponentially dismutually independent iid sequences of random vari- tributed with mean 1/#. Let p(N) and p(N + 1) deables with distribution Fi for state i. We assume note the average arrival rates for these two systems. that each source exhibits a stationary regime. While Note that this scaling ensures that they are identical, in state i, the source produces a fluid with rate ri. p(N) = p(N + 1). Let Q(t, N) and Q(t, N + 1) deWe compare the following two systems to each other: note the number of customers in the queues at time i) a single server queue fed by N sources, ii) a sin- t > 0 when they have infinite capacity. Let Q (N) = gle server queue fed by N + 1 sources where the limt--~oo Q(t, N) and Q ( N + I ) = limt~oo Q(t, N + I ) . source generation rate while in state i is scaled to These exist and are well defined provided p(N) < #. N/(N + 1)rl. In both cases, we assume that the We also are interested in the case where the queue service rate is C. Let p(N) and p(N + 1) denote has finite capacity of B. Let PL(N) and PL(N + 1) the offered loads for these two systems. Note that denote the stationary overflow probability of the two this scaling ensures that they are identical, p(N) = systems. p(N + 1). Let W(t, N) and W(t, N + 1) denote the We have the following results for these two fluid backlogs in the queues at time t > 0 when they have systems, infinite capacity. Let W(N) = limt~c~ W(t, N) and Theorem 3. Q(t, N + 1)t <icx Q(t, N)t W(N + 1) = limt_+~ W(t, N + 1). These exist and Corollary 2. Q(N + 1) <icx Q(N) are well defined provided p(N) < C. We also are and when the queue has a finite capacity B, interested in the case where the queue has finite caTheorem 4. PL(N + 1) <= PL(N). pacity of B. Let PL(N) and PL(N + 1) denote the These results can be generalized in several ways. stationary overflow probability of the two systems. First, they hold in the case that there is an addiWe have the following results for these two fluid tional source of work, independent of the sources. systems, Second, Theorem 3 can be restated in terms of the unfinished work in the queues and the service times Theorem 1. W(t, N + 1)t <~cx W(t, N)t Corollary 1. W(N + 1) <Sex W(N) permitted to be characterized by an arbitrary disand when the queue has a finite capacity B, tribution with mean 1/#. Theorem 2. PL(N + 1) < PL(N).

/lp/association-for-computing-machinery/stochastic-bounds-for-queueing-systems-with-multiple-markov-modulated-34R7bdC3Zl