Existence of a Limit Distribution in Queueing Systems with Bounded Sojourn Time

Afanas'eva, L. G.

doi:10.1137/1110066

Cited by 12 publications

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“…The papers listed above all consider diffusion control problems that admit pathwise solutions. 1 Because the solution of the diffusion control problem is nontrivial in our case, we must solve it by explicitly constructing a solution to the associated Hamilton-Jacobi-Bellman (HJB) equations using the approach in Kumar and Muthuraman [24]. We then verify that the solution to the HJB equations is indeed the optimal value function for the diffusion control problem.…”

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confidence: 99%

“…A natural way to incorporate impatience is to assume that each customer independently abandons the queue, or reneges, if his service has not begun within a certain amount of time, which could be a random variable drawn from a given distribution. Many papers have studied such queueing systems with reneging customers from a performance evaluation standpoint; see, for example, Afanas'eva [1], Baccelli et al [3], Daley [11], Garnett et al [12], Kovalenko [21], Lillo and Martin [25], Palm [29], Reed and Ward [32], Stanford [35], Ward and Glynn [37,38], and Zeltyn and Mandelbaum [41]. In particular, a lot is known about the behavior of these systems in asymptotic parameter regimes, such as the heavy-traffic regime for a queue with a single server (Ward and Glynn [37,38], Reed and Ward [32]), as well as the many-server regime (Garnett et al [12], Zeltyn and Mandelbaum [41]).…”

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confidence: 99%

“…where N is a standard Poisson process independent of A and S, and Q is the queue-length process. 1 Note that we do not mean pathwise stochastic control (for which few results exist), in which the cost function is defined in a pathwise way. 2 For convenience of notation, our formulation of the process R allows the customer in service to abandon.…”

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Asymptotically Optimal Admission Control of a Queue with Impatient Customers

Ward

Kumar

2008

Mathematics of OR

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We consider a GI/GI/1 queue with impatient customers in heavy traffic. We use the solution of an approximating singular diffusion control problem to construct an admission control policy for the queue. The approximating control problem does not admit a so-called pathwise solution. Hence, the resulting admission control policy depends on second-moment data. We prove asymptotic optimality of the constructed policy using weak-convergence methods.

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Asymptotically Optimal Admission Control of a Queue with Impatient Customers

Ward

Kumar

2008

Mathematics of OR

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“…Daley [7], [8] studied the distribution of the stationary waiting time of the GIIG/I model. (See also [1] and [2].) In this paper, we shall obtain a fairly complete solution for the GIIG/I model.…”

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The GI/G/1 queue with uniformly limited virtual waiting times; the finite dam

Minh

1980

Advances in Applied Probability

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This paper studies the GI/G/I queueing system in which no customer can stay longer than a fixed interval D. This is also a model for the dam with finite capacity, instantaneous water supply and constant release rule. Using analytical method together with the property that the queueing process 'starts anew' probabilistically whenever an arriving customer initiates a busy period, w.-obtain various transient and stationary results for the system/J3 P-'

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“…Investigations of the G1/G / 1 system with a general reneging distribution appear to have begun with a model proposed by Kovalenko (1961). Necessary and sufficient conditions for existence of a limiting distribution for the waiting time in a G1/G / 1 queue in which the customer's wait is bounded by a random variable are given in Afanas'eva (1965). Daley (1965) also gives conditions for the existence of the limiting waiting time distribution, and derives an integral equation for the distribution.…”

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On queues with impatience

Stanford

1990

Adv. Appl. Probab.

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Volume 16 (1984) of Advances in Applied Probability contains a paper entitled 'Singleserver queues with impatient customers', by Baccelli,, referred to' below as B. The authors of this paper were unaware of some research of a similar nature previously published in the literature, the most recent being 'Reneging phenomena in single channel queues', by R. E. Stanford, which appeared in Vol. 4 (May 1979) Probability and Its Applications, Vol. 10 (1965), pp. 515-522, by L. G. Afanas'eva (translated from Russian).The purpose of this note is to provide a bibliographical update on the subject of impatience in general queueing systems, and to outline some of the similarities and differences in the results from the existing papers on the subject. All of the papers mentioned here are concerned with GI/G/1 queues in which a customer drops out of the queue if his waiting time exceeds his patience time, which is a random variable.Many studies of reneging phenomena have been carried out for queueing systems with specific (eg. exponential) interarrival, service, or reneging distributions. The combined list of references from S, B, and Daley (1965) provide an extensive bibliography of this research. Investigations of the G1/G / 1 system with a general reneging distribution appear to have begun with a model proposed by Kovalenko (1961). Necessary and sufficient conditions for existence of a limiting distribution for the waiting time in a G1/G / 1 queue in which the customer's wait is bounded by a random variable are given in Afanas'eva (1965). Daley (1965) also gives conditions for the existence of the limiting waiting time distribution, and derives an integral equation for the distribution. The integral equation is solved for certain special cases previously examined in the literature.The two more recent papers Band S also contain sufficient conditions for a unique stationary distribution for the offered waiting time of arriving customers, and both contain new material concerning various characteristics of the GI/G/1 system with reneging. In some cases, the results from the two papers are identical except for notation. A partial list of the expressions in B which have a counterpart in S is given below, by description and expression number, followed by the corresponding relation number from S: definition of the offered waiting time-(2.1) in B, (7) in S; the limiting distribution of virtual waiting time-(3.6) in B, (20) in S; equations involving the probability of rejection (reneging)-(3.7) and (3.8) in B, (27) and (28) in S; a Pollaczek-Kintchine formula analog-remark #1 following (3.11) in B, (23) in S; a relationship between the limiting distribution for virtual and actual waiting time-(3.11) in B, equivalent to (21) in S; an integral equation for the distribution of waiting times-(2.3) in B, (43) in S. The approaches of Band S do differ in the manner of derivation of many of the results that they have in common. The most significant difference concerns the determination of sufficient conditions for a unique stationary dis...

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Existence of a Limit Distribution in Queueing Systems with Bounded Sojourn Time

Cited by 12 publications

References 3 publications

Asymptotically Optimal Admission Control of a Queue with Impatient Customers

Asymptotically Optimal Admission Control of a Queue with Impatient Customers

The GI/G/1 queue with uniformly limited virtual waiting times; the finite dam

On queues with impatience

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