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

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.

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

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

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

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.

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

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.

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

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.