The queueing system M<sup>X</sup>/G/1 and its ramifications

Chaudhry, M. L.

doi:10.1002/nav.3800260411

Cited by 27 publications

(21 citation statements)

References 9 publications

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“…It can be verified that these distributions agree with the results from Chaudhry [3] for the model without dependencies between successive service times.…”

Section: Poisson Batch Arrivals: Stationary Queue Length At Arrival Asupporting

confidence: 83%

A single-server queue with batch arrivals and semi-Markov services

et al. 2017

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We investigate the transient and stationary queue length distributions of a class of service systems with correlated service times. The classical M X /G/1 queue with semi-Markov service times is the most prominent example in this class and serves as a vehicle to display our results. The sequence of service times is governed by a modulating process J (t). The state of J (·) at a service initiation time determines the joint distribution of the subsequent service duration and the state of J (·) at the next service initiation. Several earlier works have imposed technical conditions, on the zeros of a matrix determinant arising in the analysis, that are required in the computation of the stationary queue length probabilities. The imposed conditions in several of these articles are difficult or impossible to verify. Without such assumptions, we determine both the transient and the steady-state joint distribution of the number of customers immediately after a departure and the state of the process J (t) at the start of the next service. We numerically investigate how the mean queue length is affected by variability in the number of customers that arrive during a single service time. Our main observations here are that increasing variability may reduce the mean

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“…It can be verified that these distributions agree with the results from Chaudhry [3] for the model without dependencies between successive service times.…”

Section: Poisson Batch Arrivals: Stationary Queue Length At Arrival Asupporting

confidence: 83%

A single-server queue with batch arrivals and semi-Markov services

et al. 2017

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“…Further, it is observed that for = 0, this reduces to the result obtained by Chaudhry [4] for a regular M X /G/1 queue.…”

Section: Mean Queue Sizementioning

confidence: 48%

“…Note that for = 0, we get the PGF of the stationary queue size distribution for a regular M X /G/1 queue (e.g., see Chaudhry [4]) from our Theorem 2. Also, we note that if we take Prob[X = 1], then E X = 1 and X z = z and our Theorem 2 agree with Equation (3) of Medhi [23].…”

Section: Analysis Of the Queue Without N -Policymentioning

confidence: 97%

“…The PGF of the queue size distribution at a departure epoch for an M x /G/1 queue was obtained by Sahbazov [26] and Chaudhry [4] through different approaches. However, in this section, we generalized their result for our M X / G 1 G 2 /1/N -policy queue, which was neither studied by Lee et al [17,18] nor by Lee and Srinivasan [16], even for the M x /G/1 type of queueing model.…”

Section: Queue Size Distribution At a Departure Epochmentioning

confidence: 99%

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A Batch Arrival Queue with a Second Optional Service Channel UnderN-Policy

Choudhury

Paul

2006

Stochastic Analysis and Applications

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We consider an M X /G/1 queueing system with a second optional service channel under N -policy. The server remains idle until the queue size reaches or exceeds N (≥1). As soon as the queue size becomes at least N , the server immediately begins to serve the first essential service to all the waiting customers. After the completion of which, only some of them receive the second optional service. For this model, our study is basically concentrated in obtaining the queue size distribution at a random epoch as well as at a departure epoch. Further, we derive a simple procedure to obtain optimal stationary policy under a suitable linear cost structure. Moreover, we provide some important performance measures of this model with some numerical examples.

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“…This work generalized models considered by various authors (see for instance, Chaudhry and Templeton [3,4]). By assuming that, in addition, the input stream is general bulk, we obviously have another problem of the cmtical behavior of the queueing process about level r. Indeed, in this case, when the server will start his service for the first time after being idle, the queue length is more likely to exceed than to exactly reach level r. The authors apply the recent results of the first passage problem obtained in Abolnikov and Dshalalow [1] to study the behavior of the queueing process at the instant of the first passage time of level r by the process.…”

Section: Introductionmentioning

confidence: 83%