Delay Moment Bounds for Multiserver Queues with Infinite Variance Service Times

The overflow process of the multiserver queue with phase-type service times and finite waiting room is a Markov renewal process. The solution for this process is obtained. If the service times are exponential the overflow process reduces to a renewal process. For the latter case explicit expressions and numerical results are given.

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The asymptotic variance of departures in critically loaded queues

Advances in Applied Probability ◽

10.1017/s0001867800004778 ◽

2011 ◽

Vol 43 (01) ◽

pp. 243-263 ◽

Cited By ~ 2

Author(s):

A. Al Hanbali ◽

M. Mandjes ◽

Y. Nazarathy ◽

W. Whitt

Keyword(s):

Explicit Expression ◽

Counting Process ◽

Asymptotic Variance ◽

Arrival Rate ◽

Finite Capacity ◽

System Load ◽

Multiserver Queues ◽

Service Patterns ◽

Coefficients Of Variation ◽

Service Times

We consider the asymptotic variance of the departure counting process D(t) of the GI/G/1 queue; D(t) denotes the number of departures up to time t. We focus on the case where the system load ϱ equals 1, and prove that the asymptotic variance rate satisfies lim t→∞varD(t) / t = λ(1–2/π)(c a 2 + c s 2), where λ is the arrival rate, and c a 2 and c s 2 are squared coefficients of variation of the interarrival and service times, respectively. As a consequence, the departures variability has a remarkable singularity in the case in which ϱ equals 1, in line with the BRAVO (balancing reduces asymptotic variance of outputs) effect which was previously encountered in finite-capacity birth-death queues. Under certain technical conditions, our result generalizes to multiserver queues, as well as to queues with more general arrival and service patterns. For the M/M/1 queue, we present an explicit expression of the variance of D(t) for any t.

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Discrete-time multiserver queues with geometric service times

Computers & Operations Research ◽

10.1016/s0305-0548(02)00155-7 ◽

2004 ◽

Vol 31 (1) ◽

pp. 81-99 ◽

Cited By ~ 36

Author(s):

Peixia Gao ◽

Sabine Wittevrongel ◽

Herwig Bruneel

Keyword(s):

Discrete Time ◽

Multiserver Queues ◽

Service Times

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The asymptotic variance of departures in critically loaded queues

Advances in Applied Probability ◽

10.1239/aap/1300198521 ◽

2011 ◽

Vol 43 (1) ◽

pp. 243-263 ◽

Cited By ~ 14

Author(s):

A. Al Hanbali ◽

M. Mandjes ◽

Y. Nazarathy ◽

W. Whitt

Keyword(s):

Explicit Expression ◽

Counting Process ◽

Asymptotic Variance ◽

Arrival Rate ◽

Finite Capacity ◽

System Load ◽

Multiserver Queues ◽

Service Patterns ◽

Coefficients Of Variation ◽

Service Times

We consider the asymptotic variance of the departure counting processD(t) of the GI/G/1 queue;D(t) denotes the number of departures up to timet. We focus on the case where the system load ϱ equals 1, and prove that the asymptotic variance rate satisfies limt→∞varD(t) /t= λ(1–2/π)(ca2+cs2), where λ is the arrival rate, andca2andcs2are squared coefficients of variation of the interarrival and service times, respectively. As a consequence, the departures variability has a remarkable singularity in the case in which ϱ equals 1, in line with the BRAVO (balancing reduces asymptotic variance of outputs) effect which was previously encountered in finite-capacity birth-death queues. Under certain technical conditions, our result generalizes to multiserver queues, as well as to queues with more general arrival and service patterns. For the M/M/1 queue, we present an explicit expression of the variance ofD(t) for anyt.

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Calculation of delay characteristics for multiserver queues with constant service times

European Journal of Operational Research ◽

10.1016/j.ejor.2008.10.017 ◽

2009 ◽

Vol 199 (1) ◽

pp. 170-175 ◽

Cited By ~ 6

Author(s):

Peixia Gao ◽

Sabine Wittevrongel ◽

Joris Walraevens ◽

Marc Moeneclaey ◽

Herwig Bruneel

Keyword(s):

Multiserver Queues ◽

Service Times ◽

Constant Service Times

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SRPT Scheduling Discipline in Many-Server Queues with Impatient Customers

Management Science ◽

10.1287/mnsc.2021.4110 ◽

2021 ◽

Author(s):

Jing Dong ◽

Rouba Ibrahim

Keyword(s):

Priority Queue ◽

System Throughput ◽

Single Server ◽

Impatient Customers ◽

Scheduling Policy ◽

Multiserver Queues ◽

Many Server Queues ◽

Service Times ◽

The Many ◽

Queues With Abandonment

The shortest-remaining-processing-time (SRPT) scheduling policy has been extensively studied, for more than 50 years, in single-server queues with infinitely patient jobs. Yet, much less is known about its performance in multiserver queues. In this paper, we present the first theoretical analysis of SRPT in multiserver queues with abandonment. In particular, we consider the [Formula: see text] queue and demonstrate that, in the many-sever overloaded regime, performance in the SRPT queue is equivalent, asymptotically in steady state, to a preemptive two-class priority queue where customers with short service times (below a threshold) are served without wait, and customers with long service times (above a threshold) eventually abandon without service. We prove that the SRPT discipline maximizes, asymptotically, the system throughput, among all scheduling disciplines. We also compare the performance of the SRPT policy to blind policies and study the effects of the patience-time and service-time distributions. This paper was accepted by Baris Ata, stochastic models & simulation.

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