A queue with poisson input and semi-Markov service times: busy period analysis

Purdue, Peter

doi:10.1017/s0021900200048051

Cited by 6 publications

(7 citation statements)

References 6 publications

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“…Although several papers have studied such a queueing model (cf. [1,6,11,15,19,25,26,27,30]), we still need to make several adjustments in order to make it applicable to our situation. The analysis below is based on the framework that was recently introduced in [1].…”

Section: Queue Length Analysismentioning

confidence: 99%

Generalized gap acceptance models for unsignalized intersections

2019

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This paper contributes to the modeling and analysis of unsignalized intersections. In classical gap acceptance models vehicles on the minor road accept any gap greater than the critical gap, and reject gaps below this threshold, where the gap is the time between two subsequent vehicles on the major road. The main contribution of this paper is to develop a series of generalizations of existing models, thus increasing the model's practical applicability significantly. First, we incorporate driver impatience behavior while allowing for a realistic merging behavior; we do so by distinguishing between the critical gap and the merging time, thus allowing multiple vehicles to use a sufficiently large gap. Incorporating this feature is particularly challenging in models with driver impatience. Secondly, we allow for multiple classes of gap acceptance behavior, enabling us to distinguish between different driver types and/or different vehicle types. Thirdly, we use the novel M X /SM2/1 queueing model, which has batch arrivals, dependent service times, and a different service-time distribution for vehicles arriving in an empty queue on the minor road (where 'service time' refers to the time required to find a sufficiently large gap). This setup facilitates the analysis of the service-time distribution of an arbitrary vehicle on the minor road and of the queue length on the minor road. In particular, we can compute the mean service time, thus enabling the evaluation of the capacity for the minor road vehicles.

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Section: Queue Length Analysismentioning

confidence: 99%

Generalized gap acceptance models for unsignalized intersections

2019

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“…For the semi-Markov model this can be established provided that the total number of types of customers is finite (as e.g. in [19], [22]); for the replacement model, provided the number of services given by each replacement is bounded (see [3] for details). Unfortunately, it appears difficult to establish it generally.…”

Section: Respectivelymentioning

confidence: 99%

“…The attractiveness of such problems stems as much from their mathematical tractability as their usefulness as models, and much work has been done on a variety of such problems. Instances, among single-server models, are the discrete Markovian environment models of Neuts [19] and others ( [10], [11], [22], [1], [2], [29]); the continuous Markovian environment models of Neuts [20] and [28], [29], [21]; and what may be described as the 'additive' environment models of Grinstein and Rubinovitch [15] and Boxma [7].…”

Section: Introductionmentioning

confidence: 99%

On queues in discrete regenerative environments, with application to the second of two queues in series

Balagopal

1979

Advances in Applied Probability

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Let Un be the time between the nth and (n + 1)th arrivals to a single-server queuing system, and Vn the nth arrival's service time. There are quite a few models in which {Un, Vn, n ≥ 1} is a regenerative sequence. In this paper, some light and heavy traffic limit theorems are proved solely under this assumption; some of the light traffic results, and all the heavy traffic results, are new for two such models treated earlier by the author; and all the results are new for the semi-Markov queuing model.In the last three sections, the results are applied to a single-server queue whose input is the output of a G/G/1 queue functioning in light traffic.

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“…Relatively little is known about queues for which interarrival times and/or service times are not independent; some studies are available however (cf. Cinlar (1967), Loynes (1962), Pearce (1967), Purdue (1975». One reason for this seems to be the lack of tractable models for dependent sequences of random variables. Recently, models have been developed for sequences of dependent exponential random variables (cf.…”

Section: Introductionmentioning

confidence: 99%

A cyclic queueing network with dependent exponential service times

Jacobs

1978

Journal of Applied Probability

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A cyclic queueing network with two servers and a finite number of customers is studied. The service times for server 1 form an earma(1,1) process (exponential mixed autoregressive moving average process both of order 1) which is a sequence of positively correlated exponential random variables; the process in general is not Markovian. The service times for the other server are independent with a common exponential distribution. Limiting results for the number of customers in queue and the virtual waiting time at server 1 are obtained. Comparisons are made with the case of independent exponential service times for server 1.

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A queue with poisson input and semi-Markov service times: busy period analysis

Cited by 6 publications

References 6 publications

Generalized gap acceptance models for unsignalized intersections

Generalized gap acceptance models for unsignalized intersections

On queues in discrete regenerative environments, with application to the second of two queues in series

A cyclic queueing network with dependent exponential service times

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