Poisson Lindley process and its main properties

2019 ◽  
Vol 152 ◽  
pp. 74-81
Author(s):  
Ji Hwan Cha
Keyword(s):  
Lindley Process

Evaluation of the total time in system in a preempt/resume priority queue via a modified Lindley process

1983 ◽  
Vol 15 (04) ◽  
pp. 840-856 ◽  
Author(s):  
J. Keilson ◽  
U. Sumita
Keyword(s):  
Time Distribution ◽  
Arrival Rate ◽  
Priority Queue ◽  
Service Time Distribution ◽  
Transform Method ◽  
Laguerre Transform ◽  
Poisson Stream ◽  
Lindley Process ◽  
Theoretical Results ◽  
Time Sequences

A Poisson stream of arrival rate λI and service-time distribution A I(x) has preempt/resume priority over a second stream of rate λII and distribution A II(x). Abundant theoretical results exist for this system, but severe numerical difficulties have made many descriptive distributions unavailable. Moreover, the distribution of total time in system of low-priority customers has not been discussed theoretically. It is shown that the waiting-time sequences of such customers before first entry into service is a Lindley process modified by replacement. This leads to the total time distribution needed. A variety of descriptive distributions, transient and stationary, is obtained numerically via the Laguerre transform method.

Transient and busy period analysis of the GIG/1 Queue as a Hilbert factorization problem

1991 ◽  
Vol 28 (4) ◽  
pp. 873-885 ◽  
Author(s):  
Dimitris J. Bertsimas ◽  
Julian Keilson ◽  
Daisuke Nakazato ◽  
Hongtao Zhang
Keyword(s):  
Closed Form ◽  
Waiting Time ◽  
Time Distribution ◽  
Busy Period ◽  
Laplace Transforms ◽  
Waiting Time Distribution ◽  
Period Analysis ◽  
Factorization Problem ◽  
Period Distribution ◽  
Lindley Process

In this paper we find the waiting time distribution in the transient domain and the busy period distribution of the GI G/1 queue. We formulate the problem as a two-dimensional Lindley process and then transform it to a Hilbert factorization problem. We achieve the solution of the factorization problem for the GI/R/1, R/G/1 queues, where R is the class of distributions with rational Laplace transforms. We obtain simple closed-form expressions for the Laplace transforms of the waiting time distribution and the busy period distribution. Furthermore, we find closed-form formulae for the first two moments of the distributions involved.

Modified Lindley process with replacement: dynamic behavior, asymptotic decomposition and applications

1989 ◽  
Vol 26 (03) ◽  
pp. 552-565 ◽  
Author(s):  
J. George Shanthikumar ◽  
Ushio Sumita
Keyword(s):  
Stochastic Process ◽  
Discrete Time ◽  
Queueing Systems ◽  
Random Variables ◽  
Decomposition Theorem ◽  
Asymptotic Decomposition ◽  
Ergodic Decomposition ◽  
Special Cases ◽  
Unified View ◽  
Lindley Process

We consider a discrete-time stochastic process {Wn , n≧0} governed by i.i.d random variables {ξ n } whose distribution has support on (–∞,∞) and replacement random variables {Rn } whose distributions have support on [0,∞). Given Wn, Wn + 1 takes the value Wn + ζ n + 1 if it is non-negative. Otherwise Wn + 1 takes the value Rn + 1 where the distribution of Rn + 1 depends only on the value of Wn + ζn + 1 . This stochastic process is reduced to the ordinary Lindley process for GI/G/1 queues when Rn = 0 and is called a modified Lindley process with replacement (MLPR). It is shown that a variety of queueing systems with server vacations or priority can be formulated as MLPR. An ergodic decomposition theorem is given which contains recent results of Doshi (1985) and Keilson and Servi (1986) as special cases, thereby providing a unified view.

Modified Lindley process with replacement: dynamic behavior, asymptotic decomposition and applications

1989 ◽  
Vol 26 (3) ◽  
pp. 552-565 ◽  
Author(s):  
J. George Shanthikumar ◽  
Ushio Sumita
Keyword(s):  
Stochastic Process ◽  
Discrete Time ◽  
Queueing Systems ◽  
Random Variables ◽  
Decomposition Theorem ◽  
Asymptotic Decomposition ◽  
Ergodic Decomposition ◽  
Special Cases ◽  
Unified View ◽  
Lindley Process

We consider a discrete-time stochastic process {Wn, n≧0} governed by i.i.d random variables {ξ n} whose distribution has support on (–∞,∞) and replacement random variables {Rn} whose distributions have support on [0,∞). Given Wn, Wn+ 1 takes the value Wn + ζ n+ 1 if it is non-negative. Otherwise Wn+ 1 takes the value Rn +1 where the distribution of Rn+ 1 depends only on the value of Wn + ζn +1. This stochastic process is reduced to the ordinary Lindley process for GI/G/1 queues when Rn = 0 and is called a modified Lindley process with replacement (MLPR). It is shown that a variety of queueing systems with server vacations or priority can be formulated as MLPR. An ergodic decomposition theorem is given which contains recent results of Doshi (1985) and Keilson and Servi (1986) as special cases, thereby providing a unified view.

Transient and busy period analysis of the GI G/1 Queue as a Hilbert factorization problem

1991 ◽  
Vol 28 (04) ◽  
pp. 873-885 ◽  
Author(s):  
Dimitris J. Bertsimas ◽  
Julian Keilson ◽  
Daisuke Nakazato ◽  
Hongtao Zhang
Keyword(s):  
Closed Form ◽  
Waiting Time ◽  
Time Distribution ◽  
Busy Period ◽  
Laplace Transforms ◽  
Waiting Time Distribution ◽  
Period Analysis ◽  
Factorization Problem ◽  
Period Distribution ◽  
Lindley Process

In this paper we find the waiting time distribution in the transient domain and the busy period distribution of the GI G/1 queue. We formulate the problem as a two-dimensional Lindley process and then transform it to a Hilbert factorization problem. We achieve the solution of the factorization problem for the GI/R/1, R/G/1 queues, where R is the class of distributions with rational Laplace transforms. We obtain simple closed-form expressions for the Laplace transforms of the waiting time distribution and the busy period distribution. Furthermore, we find closed-form formulae for the first two moments of the distributions involved.

Evaluation of the total time in system in a preempt/resume priority queue via a modified Lindley process

1983 ◽  
Vol 15 (4) ◽  
pp. 840-856 ◽  
Author(s):  
J. Keilson ◽  
U. Sumita
Keyword(s):  
Time Distribution ◽  
Arrival Rate ◽  
Priority Queue ◽  
Service Time Distribution ◽  
Transform Method ◽  
Laguerre Transform ◽  
Poisson Stream ◽  
Lindley Process ◽  
Theoretical Results ◽  
Time Sequences

A Poisson stream of arrival rate λI and service-time distribution AI(x) has preempt/resume priority over a second stream of rate λII and distribution AII(x). Abundant theoretical results exist for this system, but severe numerical difficulties have made many descriptive distributions unavailable. Moreover, the distribution of total time in system of low-priority customers has not been discussed theoretically. It is shown that the waiting-time sequences of such customers before first entry into service is a Lindley process modified by replacement. This leads to the total time distribution needed. A variety of descriptive distributions, transient and stationary, is obtained numerically via the Laguerre transform method.

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