An upper bound for multi-channel queues

Wolff, Ronald W.

doi:10.2307/3213363

Cited by 68 publications

(28 citation statements)

References 3 publications

(3 reference statements)

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“…• Remark: Our proof of Theorem 8 is closely related not only to Wolff's proof, 9 but also to Sonderman's comparison proofs. 2 0 , 2 1 Sonderman was concerned with the effect of different service-time distributions instead of different queue disciplines.…”

Section: T O Continue the Induction Proof For B N Note That (10) Anmentioning

confidence: 83%

“…9 As before, we assume the FCFS discipline, but now we allow the arrival streams in the two separate systems to be arbitrary. W e assume the service times are independent of the arrival processes and mutually independent and identically distributed, but they need not be exponentially distributed.…”

Section: Let {X(t) T > 0} and {Y(t) T > 0} Be Real-valued Stochastimentioning

confidence: 99%

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Resource Sharing for Efficiency in Traffic Systems

Smith¹,

Whitt²

1981

Bell System Technical Journal

162

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Experience has shown that efficiency usually increases when sep arate traffic systems are combined into a single system. For example, if Group A contains 10 trunks and Group Β 8 trunks, there should be fewer blocked calls if A and Β are combined into a single group of 18 trunks. It is intuitively clear that the separate systems are less efficient because a call can be blocked in one when trunks are idle in the other. Teletraffic engineers and queuing theorists widely accept such efficiency principles and often assume that their mathematical proofs are either trivial or already in the literature. This is not the case for two fundamental problems that concern combining blocking systems (as in the example above) and combining delay systems. For the simplest models, each problem reduces to the proof of an inequality involving the corresponding classical Erlang function. Here the two inequalities are proved in two different ways by exploiting general stochastic comparison concepts: first, by monotone likelihood-ratio methods and, second, by sample-path or "coupling" methods. These methods not only yield the desired inequalities and stronger compar isons for the simplest models, but also apply to general arrival processes and general service-time distributions. However, it is as sumed that the service-time distributions are the same in the systems being combined. This common-distribution condition is crucial since it may be disadvantageous to combine systems with different servicetime distributions. For instance, the adverse effect of infrequent long calls in one system on frequent short calls in the other system can outweigh the benefits of making the two groups of servers mutually accessible. I. I N T R O D U C T I O N A N D S U M M A R YFrom extensive experience in teletraffic engineering, it is well known that congestion can often be reduced by sharing resources. The block- 9ing probability in a loss system and the average waiting time in a delay system are usually much less when separate facilities serving separate streams of traffic are combined to serve all the streams together. Alternatively, for a given level of congestion, fewer facilities are usually required to serve the streams together. Sometimes such results are trivial: Whenever the combined system may be managed as if it were in fact separate systems, the optimal performance of the combined system is at least as good as that of the separate systems. However, such management is not allowed in the models treated here. In any case, the efficiency of shared resources is certainly a fundamental principle of teletraffic engineering. The purpose of this paper is to establish versions of this efficiency principle mathematically. Our first two results verify conjectures by Arthurs and Stuck. 1 T o state our first result, let L(s, λ μ) denote the stationary loss or overflow rate in an M/M/s loss system (no waiting room) with s servers, arrival rate λ, and individual service rate μ. (See Kleinrock 2 for background on the queuing models.) It is well known that L(s...

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Section: T O Continue the Induction Proof For B N Note That (10) Anmentioning

confidence: 83%

Section: Let {X(t) T > 0} and {Y(t) T > 0} Be Real-valued Stochastimentioning

confidence: 99%

Resource Sharing for Efficiency in Traffic Systems

Smith¹,

Whitt²

1981

Bell System Technical Journal

162

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“…If anticipative information (on processing times for future arrivals) is not permitted, the centralized controller has the most possible information when it knows the processing times of all tasks upon arrival. In this case, routing tasks to the processor with the least work, which is equivalent to FCFS with a shared queue, has been shown to be optimal for various objectives (Kingman, 1970, Vasicek, 1977, Foss, 1980and 1981, Wol , 1977, Daley, 1987, Liu and Towsley, 1994band 1994c.…”

Section: Inriamentioning

confidence: 99%

Optimal Load Balancing on Distributed Homogeneous Unreliable Processors

Liu

Righter

1998

Operations Research

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We consider optimal load balancing in a distributed computing environment with several homogeneous unreliable processors that have limited state information. Each processor receives its own arrival process of tasks from outside users, some of which can be redirected to the other processors. Processing times are iid and arbitrarily distributed. The arrival process of outside tasks to each processor may be arbitrary as long as it is independent of the state of the system. Processors may fail, with arbitrary failure and repair processes that are also independent of the state of the system. The only information available to a processor is the history of its decisions for routing work to other processors, and the arrival times for its own arrival process. We show that the round-robin policy, in which each processor sends the tasks that can be redirected to each of the processors in turn, stochastically minimizes the n th task completion time for all n, and minimizes response times and queue lengths in a separable increasing convex sense, among all policies that balance workload. We also show that if there is a single centralized controller, round-robin is the optimal policy, and a single controller using round-robin routing is better than the optimal distributed system, where \optimal" and \better" are in the sense of stochastically minimizing task completion times and minimizing response times and queue lengths in the separable increasing convex sense.

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“…Brumelle (1971) bounded the expected waiting time. Wolff (1977) bounded moments of the delay distribution (see also Wolff (1987) for corrections and comments). Halfin (1985) bounded the probability distribution of the number of customers in the system, and its expected value in equilibrium.…”

Section: Joining the Shortest Queuementioning

confidence: 99%

Bounding functions of Markov processes and the shortest queue problem

Gubner¹,

Gopinath²,

Varadhan³

1989

Advances in Applied Probability

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We prove a theorem which can be used to show that the expectation of a non-negative function of the state of a time-homogeneous Markov process is uniformly bounded in time. This is reminiscent of the classical theory of non-negative supermartingales, except that our analog of the supermartingale inequality need not hold almost surely. Consequently, the theorem is suitable for establishing the stability of systems that evolve in a stabilizing mode in most states, though from certain states they may jump to a less stable state. We use this theorem to show that ‘joining the shortest queue' can bound the expected sum of the squares of the differences between all pairs among N queues, even under arbitrarily heavy traffic.

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An upper bound for multi-channel queues

Cited by 68 publications

References 3 publications

Resource Sharing for Efficiency in Traffic Systems

Resource Sharing for Efficiency in Traffic Systems

Optimal Load Balancing on Distributed Homogeneous Unreliable Processors

Bounding functions of Markov processes and the shortest queue problem

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