Discrete-time multiserver queues with geometric service times

A discrete time queueing model is considered to estimate of the number of customers in the system. The arrivals, which are in groups of size X, inter-arrivals times and service times are distributed independent. The inter-arrivals fallows geometric distribution with parameter p and service times follows general distribution with parameter µ, we have derive the various transient state solution along with their moments and numerical illustrations in this paper.

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Overflow processes for multiserver queues with finite waiting room

Advances in Applied Probability ◽

10.1017/s0001867800021881 ◽

1984 ◽

Vol 16 (1) ◽

pp. 8-8

Author(s):

Jos H. A. De Smit

Keyword(s):

Renewal Process ◽

Numerical Results ◽

Markov Renewal Process ◽

Waiting Room ◽

Multiserver Queues ◽

Phase Type ◽

Service Times ◽

Overflow Process ◽

Explicit Expressions

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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DISCRETE-TIME GIX/GEO/1/N QUEUE WITH WORKING VACATIONS AND VACATION INTERRUPTION

Asia Pacific Journal of Operational Research ◽

10.1142/s0217595914500031 ◽

2014 ◽

Vol 31 (01) ◽

pp. 1450003 ◽

Cited By ~ 3

Author(s):

SHAN GAO ◽

ZAIMING LIU ◽

QIWEN DU

Keyword(s):

Discrete Time ◽

Finite Buffer ◽

Waiting Time Distribution ◽

Service Period ◽

Imbedded Markov Chain ◽

Working Vacations ◽

Vacation Interruption ◽

Service Times ◽

System Length ◽

Vacation Period

In this paper, we study a discrete-time finite buffer batch arrival queue with multiple geometric working vacations and vacation interruption: the server serves the customers at the lower rate rather than completely stopping during the vacation period and can come back to the normal working level once there are customers after a service completion during the vacation period, i.e., a vacation interruption happens. The service times during a service period, service times during a vacation period and vacation times are geometrically distributed. The queue is analyzed using the supplementary variable and the imbedded Markov-chain techniques. We obtain steady-state system length distributions at pre-arrival, arbitrary and outside observer's observation epochs. We also present probability generation function (p.g.f.) of actual waiting-time distribution in the system and some performance measures.

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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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