A queue with Markov-dependent service times

A queueing system is considered where each item has a property associated with it, and where the service time interposed between two items depends on the properties of both of these items. The steady state of a single-channel queue of this type, with Poisson input, is investigated. It is shown how the probability generating function of the number of items waiting can be found. Easily applied approximations are given for the mean number of items waiting and for the average waiting time.

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On the waiting time distribution in a generalized queueing system with uniformly bounded sojourn times

Journal of Applied Probability ◽

10.2307/3212332 ◽

1972 ◽

Vol 9 (3) ◽

pp. 642-649 ◽

Cited By ~ 8

Author(s):

Jacqueline Loris-Teghem

Keyword(s):

Generating Function ◽

Waiting Time ◽

Service Time ◽

Time Distribution ◽

Queueing System ◽

Waiting Time Distribution ◽

Stieltjes Transform ◽

Sojourn Times ◽

Stieltjes Transforms ◽

Uniformly Bounded

A generalized queueing system with (N + 2) types of triplets (delay, service time, probability of joining the queue) and with uniformly bounded sojourn times is considered. An expression for the generating function of the Laplace-Stieltjes transforms of the waiting time distributions is derived analytically, in a case where some of the random variables defining the model have a rational Laplace-Stieltjes transform.The standard Kl/Km/1 queueing system with uniformly bounded sojourn times is considered in particular.

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How many random digits are required until given sequences are obtained?

Journal of Applied Probability ◽

10.1017/s0021900200037025 ◽

1982 ◽

Vol 19 (03) ◽

pp. 518-531 ◽

Cited By ~ 12

Author(s):

Gunnar Blom ◽

Daniel Thorburn

Keyword(s):

Generating Function ◽

Waiting Time ◽

Probability Generating Function ◽

Recursive Formula ◽

Fixed Number ◽

Digit Sequence ◽

Mean Waiting Time ◽

The Mean ◽

The Given

Random digits are collected one at a time until a given k -digit sequence is obtained, or, more generally, until one of several k -digit sequences is obtained. In the former case, a recursive formula is given, which determines the distribution of the waiting time until the sequence is obtained and leads to an expression for the probability generating function. In the latter case, the mean waiting time is given until one of the given sequences is obtained, or, more generally, until a fixed number of sequences have been obtained, either different sequences or not necessarily different ones. Several results are known before, but the methods of proof seem to be new.

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Supplementary variable method applied to the MAP/G/1 queueing system

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s033427000001239x ◽

1998 ◽

Vol 40 (1) ◽

pp. 86-96 ◽

Cited By ~ 12

Author(s):

Bong Dae Choi ◽

Gang Uk Hwang ◽

Dong Hwan Han

Keyword(s):

Steady State ◽

Generating Function ◽

Service Time ◽

Queue Length ◽

Queueing System ◽

Variable Method ◽

Supplementary Variable ◽

Supplementary Variable Method ◽

One Dimensional

AbstractIn this paper we consider the MAP/G/1 queueing system with infinite capacity. In analysis, we use the supplementary variable method to derive the double transform of the queue length and the remaining service time of the customer in service (if any) in the steady state. As will be shown in this paper, our method is very simple and elegant. As a one-dimensional marginal transform of the double transform, we obtain the generating function of the queue length in the system for the MAP/G/1 queue, which is consistent with the known result.

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On a non-Markovian time-dependent queueing system

Journal of Applied Probability ◽

10.2307/3214324 ◽

1989 ◽

Vol 26 (1) ◽

pp. 142-151 ◽

Cited By ~ 2

Author(s):

S. D. Sharma

Keyword(s):

Steady State ◽

Generating Function ◽

Queue Length ◽

Queueing System ◽

Length Distribution ◽

Probability Generating Function ◽

Transient State ◽

Time Dependent ◽

Queue Length Distribution ◽

State Behaviour

This paper studies the transient and steady-state behaviour of a continuous and discrete-time queueing system with non-Markovian type of departure mechanism. The Laplace transforms of the probability generating function of the time-dependent queue length distribution in the transient state are obtained and the probability generating function of the queue length distribution in the steady state is derived therefrom. Finally, some particular cases are discussed.

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On a non-Markovian time-dependent queueing system

Journal of Applied Probability ◽

10.1017/s0021900200041875 ◽

1989 ◽

Vol 26 (01) ◽

pp. 142-151

Author(s):

S. D. Sharma

Keyword(s):

Steady State ◽

Generating Function ◽

Queue Length ◽

Queueing System ◽

Length Distribution ◽

Probability Generating Function ◽

Transient State ◽

Time Dependent ◽

Queue Length Distribution ◽

State Behaviour

This paper studies the transient and steady-state behaviour of a continuous and discrete-time queueing system with non-Markovian type of departure mechanism. The Laplace transforms of the probability generating function of the time-dependent queue length distribution in the transient state are obtained and the probability generating function of the queue length distribution in the steady state is derived therefrom. Finally, some particular cases are discussed.

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Reservicing some customers inM/G/1queues under three disciplines

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171204210389 ◽

2004 ◽

Vol 2004 (32) ◽

pp. 1715-1723 ◽

Cited By ~ 2

Author(s):

M. R. Salehi-Rad ◽

K. Mengersen ◽

G. H. Shahkar

Keyword(s):

Steady State ◽

Generating Function ◽

Production Line ◽

Busy Period ◽

Probability Generating Function ◽

Idle Period ◽

The Mean ◽

The Moment ◽

State Size

Consider anM/G/1production line in which a production item is failed with some probability and is then repaired. We consider three repair disciplines depending on whether the failed item is repaired immediately or first stockpiled and repaired after all customers in the main queue are served or the stockpile reaches a specified threshold. For each discipline, we find the probability generating function (p.g.f.) of the steady-state size of the system at the moment of departure of the customer in the main queue, the mean busy period, and the probability of the idle period.

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On the waiting time distribution in a generalized queueing system with uniformly bounded sojourn times

Journal of Applied Probability ◽

10.1017/s0021900200035932 ◽

1972 ◽

Vol 9 (03) ◽

pp. 642-649

Author(s):

Jacqueline Loris-Teghem

Keyword(s):

Generating Function ◽

Waiting Time ◽

Service Time ◽

Time Distribution ◽

Queueing System ◽

Waiting Time Distribution ◽

Stieltjes Transform ◽

Sojourn Times ◽

Stieltjes Transforms ◽

Uniformly Bounded

A generalized queueing system with (N+ 2) types of triplets (delay, service time, probability of joining the queue) and with uniformly bounded sojourn times is considered. An expression for the generating function of the Laplace-Stieltjes transforms of the waiting time distributions is derived analytically, in a case where some of the random variables defining the model have a rational Laplace-Stieltjes transform.The standardKl/Km/1 queueing system with uniformly bounded sojourn times is considered in particular.

Download Full-text

How many random digits are required until given sequences are obtained?

Journal of Applied Probability ◽

10.2307/3213511 ◽

1982 ◽

Vol 19 (3) ◽

pp. 518-531 ◽

Cited By ~ 53

Author(s):

Gunnar Blom ◽

Daniel Thorburn

Keyword(s):

Generating Function ◽

Waiting Time ◽

Probability Generating Function ◽

Recursive Formula ◽

Fixed Number ◽

Digit Sequence ◽

Mean Waiting Time ◽

The Mean ◽

The Given

Random digits are collected one at a time until a given k -digit sequence is obtained, or, more generally, until one of several k -digit sequences is obtained. In the former case, a recursive formula is given, which determines the distribution of the waiting time until the sequence is obtained and leads to an expression for the probability generating function. In the latter case, the mean waiting time is given until one of the given sequences is obtained, or, more generally, until a fixed number of sequences have been obtained, either different sequences or not necessarily different ones. Several results are known before, but the methods of proof seem to be new.

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The M/M/NRepairable Queueing System with Variable Breakdown Rates

Discrete Dynamics in Nature and Society ◽

10.1155/2013/173407 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10

Author(s):

Shengli Lv ◽

Jingbo Li

Keyword(s):

Steady State ◽

Queueing System ◽

Probability Generating Function ◽

Idle Time ◽

Transform Method ◽

Busy Time ◽

Breakdown Rates ◽

The Mean ◽

System States ◽

Steady State Probabilities

This paper considers the M/M/Nrepairable queuing system. The customers' arrival is a Poisson process. The servers are subject to breakdown according to Poisson processes with different rates in idle time and busy time, respectively. The breakdown servers are repaired by repairmen, and the repair time is an exponential distribution. Using probability generating function and transform method, we obtain the steady-state probabilities of the system states, the steady-state availability of the servers, and the mean queueing length of the model.

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