Operations Research Letters. Instability of FIFO in a simple queueing system with arbitrarily low loads

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1 Operations Research Letters 37 (2009) Contents lists available at ScienceDirect Operations Research Letters journal homepage: Instability of FIFO in a simple queueing system with arbitrarily low loads Tolga Tezcan Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, 117 Transportation Bldg, Urbana, IL 61801, United States article info abstract Article history: Received 21 October 2008 Accepted 6 May 2009 Available online 23 May 2009 We show, using a simple example, that the First-In-First-Out (FIFO) policy can be unstable in a system with arbitrarily low load. Our proof is based on the observation that the special structure of the example we use allows us to establish stability using a much simpler queueing system. Published by Elsevier B.V. Keywords: FIFO Queueing systems Stability Fluid models 1. Introduction It is now well known that a system that has theoretically sufficient capacity to meet all the demand is not necessarily stable [1,2]. There is a vast literature on how to establish stability of queueing systems using fluid modes or drift rate arguments; see [3,4]. These tools have been successfully used to establish stability of queueing systems in many different settings. Stability of FIFO has also been studied extensively; see [5 7] and the references therein. It has been shown in [5] for a stochastic queueing network and in [7] for an adversarial queue that FIFO can be unstable in arbitrarily low loads. In this paper, we prove the instability of FIFO in arbitrarily low loads using a very simple example compared to those in [5,7]. Consider a queueing system that consists of two job classes and two server pools, see Fig. 1(a). We refer to these systems as X- systems. Let j denote the arrival rate to class j. Let µ denote the service rate of a class j job by server i for i, j = 1, 2. We assume that µ > 0 for all i, j = 1, 2. Also assume that service times and interarrival times are exponential. Define (µ, ) = µ 11 µ 12 1µ µ 11 + µ 21 µ 22 1µ µ 21. (1.1) Our main result (Theorem 2.1) is that if (µ, ) > 1 the system is positive Harris recurrent (see, [3]), i.e., the underlying Markov process has a stationary distribution, and it is rate stable if (µ, ) = 1 and transient if (µ, ) < 1. From (1.1), it is not difficult to see that FIFO can be unstable even when the load on the system is very small. For example, let 1 = 2 = 1, µ 11 = µ 22 = 1000 and µ 21 = µ 12 = 0.1 so that (µ, ) 0.2, address: ttezcan@uiuc.edu. hence the system is unstable. We first note that as long as server 1 is not allowed to serve class 2 jobs and server 2 is not allowed to serve class 1 jobs, the system will be stable and the utilization of the servers will be 0.01%. It is not difficult to see that the system will be unstable as long as µ 21 < 0.5 and µ 12 < 0.5 no matter how large µ 11 and µ 22 are. To illustrate, we simulate this system with µ 21 = µ 12 = 0.49 and µ 21 = µ 12 = 0.51 with µ 11 = µ 22 = The result are shown in Fig. 1(b). The increasing curve is the number of jobs in the first queue when service rate is µ 21 = µ 12 = 0.49 and the line on the bottom is for the system with µ 21 = µ 12 = The number of jobs in the second queue, which is not plotted, exhibits a similar trend. The idea of the proof of our main result is based on the fact that, when there are jobs in a queue waiting, the class of the job a server will serve next does not depend on the system history. For instance, in the numerical example we gave in the previous paragraph, the probability that the next job belongs to either class is 50% and this is independent from the state of the system. Given this fact, it is not difficult to see that the system is unstable, since the average time server j takes to finish service is 0.5/µ j /µ j2. Again, if µ 12 = µ 21 < 0.5, this implies on average the service time of a job from either server is greater than 1. Since the total arrival rate is 2, the queue lengths are bound to explode. In general, the analysis of the stability of FIFO is very complicated. We prove the result by showing that the X-model under FIFO is equivalent to another system with a single queue. The stability of this new system can be established using traditional fluid models; see [8]. In [5], a similar result in a queuing system, where jobs visit each server several times and service time depends on the number of the visit, has been shown. However, unlike our example, the example used there has more and more servers as the load gets smaller and smaller. In [7], an adversarial queue in a system whose graph has to have at least a diameter O(r 1 log(1/r) 3 ) if the arrival /$ see front matter. Published by Elsevier B.V. doi: /j.orl

2 T. Tezcan / Operations Research Letters 37 (2009) (a) An X-model queueing system. (b) Simulation results for two systems. Fig. 1. X-model and simulation results. rate is r has been used to show instability of FIFO. Compared to these systems, our example is very simple. This is due to the fact that they only consider systems where each job class can only be served by a unique server. Allowing all servers to serve all the job classes simplifies the proof considerably. Such simplification is not possible, even when we add another server that can only serve one of the classes. However, it is possible to extend our result to the case with several servers and several job classes as long as all the servers can handle all the job classes. It should be clear from our example above that the main reason why FIFO is not stable is because servers use activities, job classserver matchings, that are not very efficient. It is common in the literature to devise scheduling policies assuming which activities are efficient; see [9,10] among others. This is accomplished by formulating a static planning problem (SPP) (see Section 2) and determining from its solution which activities should be used, called the basic activities, and which activities should not be used, called the non-basic activities. However, the SPP requires information about arrival rates, which is not always available. We note, however, that there are policies such as max-weight; see [11] and the references therein, that do not require the information about which activities are basic. The rest of this paper is organized as follows. In Section 2 we present the details of the queueing system and our main result. In Section 3 we prove the main result. 2. Queuing model and main result Consider the X-model introduced in the previous section. Assume that servers are dispatched according to a FIFO policy; when a server finishes service, that server starts serving the longest waiting job in the system if there are any. How a job is routed to servers when there is an arrival to an empty system does not matter. For concreteness, it can be assumed that a server is picked randomly with equal probability. Service times depend both on the class and the server providing service. We assume that interarrival times and service times are exponentially distributed. Let µ denote the service rate of a class j job by server i for i, j = 1, 2. We assume that µ > 0 for all i, j = 1, 2. The arrival rate to class j is denoted by j. In order to define the load on the system, it is customary to formulate a linear program that is known as the static planning problem (SPP). The SPP in this setting is defined by min s.t. µ x = j, for j = 1, 2, x apple, for i = 1, 2, x 0, for j = 1, 2 and i = 1, 2. The quantity x can be thought of as the long-run proportion of time server i serves class j jobs. The objective of the SPP is to minimize the nominal utilization of the busiest server pool. Let (, x ) be an optimal solution to the SPP. If apple 1 then theoretically, the system can be made stable under some policy. Hence, we call the load on the system. For the example we used in Section 1, the optimal solution to SPP is = 0.1%, x = 11 x = , x = 12 x 21 = 0. To illustrate the main claim of this paper, let µ 12 = µ 21, µ 11 = µ 22 > 1 = 2 = 1, and µ 12 < µ 11, then = 1/µ 11. As discussed in Section 1, as long as µ 21 < 0.5, the system is unstable no matter what the value of µ 11 is. Hence, although the system is unstable, the load on the X-model can be made arbitrarily low by choosing µ 11 arbitrarily large. Next, we give precise definitions of stability and instability before we present our main result. Let Q j (t) denote the number of class j jobs in queue and Q (t) = (Q 1 (t), Q 2 (t)). A queueing network is said to be rate stable if for each initial fixed data kq (t)k ) 0 as t!1, t with probability one, where k kis the max norm. A queueing system is said to be positive Harris recurrent if the underlying Markov process possesses a unique stationary distribution. A queueing system is said to be transient if the underlying Markov process does not admit a stationary distribution. Next we present our main result. Theorem 2.1. Consider an X-model system operating under FIFO. (a) It is positive Harris recurrent if (µ, ) > 1. (b) It is rate stable if (µ, ) 1. (c) It is transient if (µ, ) < 1. Remark 2.2. The proof of Theorem 2.1 below implies that, if µ 11 = µ 22 and µ 12 = µ 21, an X-model under FIFO has the same finite dimensional distributions as an M/G/2 system with service rate of each server equal to µ = ( ) (µ, )/2. Since an M/G/2 system can only be stable if 2 µ > ( ), this proves Theorem 2.1 in this special case.

3 314 T. Tezcan / Operations Research Letters 37 (2009) Before we proceed with the proof, we describe the assumptions on the initial conditions. Let Q (0) = (Q 1 (0), Q 2 (0)) denote the number of jobs in queues at time 0. We assume that jobs which are present in the system initially have priority over those who arrive exogenously. In which order initially present jobs are served has no effect on the stability of the system we are considering, but we assume that there is a fixed ordering and jobs are indexed according to this order from Q 1 (0) Q 2 (0)+1 to zero. We assume that if there are initially some jobs in the system, the job with the lowest index is served by server one and the second job (if there is one) is served by the second server. Without loss of generality, we can assume that initially present jobs have the same service time distribution with those arriving at the system exogenously [12]. We also assume that the initial state of the system is independent from the interarrival times and service times of externally arriving jobs. We model the service times as follows. We assume that with the nth job there is an associated uniformly distributed random variable (r.v.) u n, for n = Q 1 (0) Q 2 (0) + 1, Q 1 (0) Q 2 (0) + 2,... The service time of the nth job depends on its class and which server provides the service. If the nth job is a class j job and served by service pool i its service time is given by F 1 (u n ), where F (x) = 1 e µ x, x 0, is the cdf of an exponential distribution with rate µ. Proof of Theorem 2.1. We prove the result by constructing a single queue system that can be analyzed using traditional fluid limits. We show that this new system is equivalent to the X-model under FIFO. We first explain this new queueing system. Let Z i (t) = j, j = 1, 2, if the class of the job server i is serving at time t is j, and Z i (t) = 0 if it is idle at time t and Z(t) = (Z 1 (t), Z 2 (t)). Let X = (Q, Z). We next argue that we can actually characterize the X-model under FIFO as a single queue system. Consider the following system. There is only one queue that all arriving jobs join when both servers are busy. The arrivals to this system occur according to a Poisson process with rate Assume that the initial state of the system is identical to that of the X-model. Also, with each job there are two associated r.v. s, (u n, n ), where, as in the X-system, u n is a uniform r.v. on [0, 1], and n is a Bernoulli r.v. with P { n = 1} = 1 /( ) and P { n = 2} = 2 /( ). The nth arriving job is designated to be a class 1 job if n = 1 and a class 2 job otherwise. When a server idles to find jobs waiting in queue, that server serves the longest waiting job in the queue. We assume that {u n } and { n } are sequence of i.i.d. r.v. s. We denote the performance processes of this system by X = ( Q, Z), where Q (t) = ( Q1 (t), Q2 (t)) and Qj (t) denotes the number of class j jobs in queue at time t and Z has the same interpretation as Z. Since jobs arrive to the system according to a Poisson process, it is easily verified that X has the same finite dimensional distributions as X. Now consider the following system. Instead of assigning the class of a job at the time of arrival, assume that the class is assigned when the job leaves the queue for service. Assume that the arrivals to this system follow the same Poisson process with rate as in the previous system. Assume also that the state of the system at time zero is identical to that of the X-model. The class and the service time of a job that arrives to the system exogenously are determined by the same sequences we used above; {u n } and { n }. Specifically, the class of the k + Q 1 (0) + Q 2 (0)th job which started service is determined by the random variable k and the service time is determined conditional on the class by u n and which server is handling this job, as explained above. Jobs in the queue are served in FIFO manner. Denote the state of this system by X = (Q 0, Z 0 ), where Q 0 and Z 0 has the same interpretation as above. We refer to this system as the single queue FIFO system. 0 Now consider the X = ( Q 0, Z), where Q 0 (t) = Q1 (t) + Q2 (t). We claim that n o P X 0 (t) = X 0 (t);80 apple t < 1 = 1. First, since jobs that are initially present in the system are ordered in the same manner and have the same classes and service requirements, until the 1st externally arriving job to the system starts service, these systems are identical. For the 1st externally arriving job, the class of the job is determined by the random variable 1 and the service time by the random variable u 1 in both systems. Hence, the systems are again identical until the 2nd externally arriving job starts service. But similarly, the 2nd job also has the same characteristics in both systems. So X = X 0 a.s. Therefore, the stability of X-model under FIFO can be characterized from the stability of the single queue FIFO model. Hence, the result follows from the following proposition. Proposition 2.3. The single queue FIFO model is (a) positive Harris recurrent if (µ, ) > 1, (b) rate stable if (µ, ) 1, (c) transient if (µ, ) < 1. We prove this result in the next section. 3. Proof of Proposition 2.3 For the rest of the proof we drop 0 from the notation for simplicity. We can prove Proposition 2.3 using the traditional fluid model analysis; see [8]. Note that, for stability analysis, we can assume in this system that there is only one job class and the service times in server i has exponential distribution with rate µ with probability j /( ), for j = 1, 2 and i = 1, 2. Therefore, the average service time of a job in server i is m i = j µ 1 Allow A(t) to denote the number of arrivals to class j by time t, respectively. Let T (t) and D (t) denote the time server i allocated to class j and number of class j jobs served by server i by time t. Let S denote a Poison process with rate µ. Let X = (Q i, T, D ; i, j = 1). Note that these processes satisfy the following equations; Q (t) = Q (0) + A(t) D (t) = S T (t), D (t), T is non-decreasing, Z! t X Q (s)d T (s) = 0 for i, j = 1, 2. 0 Let i,j X r (t) = X(rt)/r. By Theorem 4.1 in [8], { X r } is tight a.s. if Q r is bounded a.s. Every limit of a subsequence of { X r } is called a fluid limit. Fluid limits satisfy a set of equations known as the fluid model equations. The stability of a queueing system can be established by considering the stability of the solutions of fluid model equations; see [8,3]. A point t is called a regular point of a function f : R +! R N for an integer N, if f is differentiable at t. The derivative of f is denoted by ḟ.

4 T. Tezcan / Operations Research Letters 37 (2009) Proposition 3.1. Fluid limits of the single queue FIFO model satisfy the following equations. Q (t) = Q (0) D (t) + ( )t, (3.1) D (t) = µ T (t), for i, j = 1, 2, (3.2) 1 D (t) = ( ) (µ, ), if Q (t) >0, (3.3) D i1 (t) D i2 (t) =, for t t o for i = 1,,2 (3.4) 2 for t o = Q (0) mini,,2 µ, where is defined as in (1.1), for a regular point t of X. Proof. Let { X r } be a sequence of fluid scaled single queue FIFO models. Assume that Q r (0) is bounded a.s. Equations (3.1) and (3.2) are proved as in [8], we next prove that every fluid limit satisfies (3.3). Assume that Q (t) >0 for some t at which X is differentiable. Clearly, X is Lipschitz and so differentiable a.e. Since Q is continuous, there exists > 0 such that Q (s) > 0, for s 2 r [t, t + ]. Assume that X! X ar r!1 u.o.c. Then, for large r we can assume that Q r (s) >0, for s 2[t, t + ]. Hence, X,2 T r (t + ) Therefore, X T (t) = 1, for i = 1, 2.,2 r T (t ) = 2, for i = 1, 2. P P But from Theorem 4.1 in [8],,2 D (t) = i,2 T (t), where i = m 1 i This proves (3.3). Next we prove that every fluid limit satisfies (3.4). Let r denote the first time point in the rth system when all the jobs initially present in the system leave the queue. Then, r apple QX r (0) F 1 min (u i), where F min is the cdf of an exponential distribution with rate µ min = min i,,2 µ. By the strong law of large numbers, we have lim sup r 1 r apple µ min Q (0) r!1 a.s. We set t 0 = µ min Q (0) for the rest of the proof. Let B r (t) denote the number of class j jobs who started service in pool i by time t. Also, let D r i (t) denote the total number of service completions by server i by time t. Note that, for i = 1, 2 B r i1 (t) Br i1 ( r ) = D r i X (t) 1 i=d r i ( r )+1 j (i), (3.5) where j (i) = 1 if the ith job which enters service in server j is a class 1 job and = 2 otherwise. Also, { j } is a sequence of independent random variables and the sequences 1 and 2 are independent. Let Br (t) = Br (rt)/r and f (s : t) = f (t ^ s) f (s). By (3.5), for any T > 0 B r (t 0 : t) a.s. as r!1. i D (t 0 : t) T! 0 (3.6) Note also that for any s > r D r (s : t) + 1/r Br (s : t) Dr (s : t) 1/r. (3.7) We have from (3.6) and (3.7) that, for any T > 0, and s > t 0 k 2 Dr i1 (s : t) 1 D r i2 (s : t)k T! 0 a.s. as r!1, for i = 1, 2. This proves (3.4). Next, we prove the stability region of the fluid models of single queue FIFO systems. We use the stability of fluid models as defined in Section 2.4 of [3]. Proposition 2.3 follows from the following result and Theorem 4.1 in [8], Corollary in [3], and Theorem 3.2 in [13]. Proposition 3.2. The fluid model of the single queue FIFO model is (a) is stable if (µ, ) > 1, (b) is weakly stable if (µ, ) 1, (c) is weakly unstable if (µ, ) < 1. Proof. To prove (i), we need to show that if Q (0) apple 1, there exists t 0 > 0 such that for t t 0, Q (t) = 0. Assume that Q (t) >0. Then, by (3.1) and (3.3), Q (t) = ( ) (µ, ) + ( ). Hence, when (µ, ) > 1, Q (t) < for some >0. The result follows from Lemma of [3]. To prove (ii), we need to show that if Q (0) = 0, Q (t) = 0 for t 0. The result follows similarly to part (i). Whenever Q (t) >0 by (3.1) and (3.3), Q (t) apple 0. Hence, if Q (0) = 0, Q (t) = 0 for t 0, since it is differentiable a.e. To prove (iii), we need to show that if Q (0) = 0, Q ( ) > 0 for some >0. First note that T i1 (t) + T i2 (t) apple 1. (3.8) Also, by (3.2) and (3.4), T i1 (t) = µ i2 1 µ i1 2 T i2 (t). Hence, again using (3.2) and (3.8) D (t) apple (µ, )( ). By (3.1) Q (t) (1 (µ, )) ( ). Since (µ, ) < 1, Q (t) > for some >0. Hence for any >0, since Q is differentiable a.e. Q ( ) >, concluding the proof. Acknowledgment The author would like to thank the anonymous referee for the suggestion on how to prove the main result of this paper using a simpler argument than the original one which appeared in the first version. References [1] A. Rybko, A. Stolyar, Ergodicity of stochastic processes describing the operation of open queueing networks, Problems of Information Transmission 28 (3) (1992) [2] S. Lu, P. Kumar, Distributed scheduling based on due dates and buffer priorities, IEEE Transactions on Automatic Control 36 (12) (1991)

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