scholarly journals The use of Spitzer's identity in the investigation of the busy period and other quantities in the queue GI/G/1

1962 ◽  
Vol 2 (3) ◽  
pp. 345-356 ◽  
Author(s):  
J. F. C. Kingmán
Keyword(s):  
Random Walk ◽  
Characteristic Function ◽  
Waiting Time ◽  
Time Distribution ◽  
Busy Period ◽  
Waiting Time Distribution ◽  
Period Distribution ◽  
Stationary Waiting Time

As an illustration of the use of his identity [10], Spitzer [11] obtained the Pollaczek-Khintchine formula for the waiting time distribution of the queue M/G/1. The present paper develops this approach, using a generalised form of Spitzer's identity applied to a three-demensional random walk. This yields a number of results for the general queue GI/G/1, including Smith' solution for the stationary waiting time, which is established under less restrictive conditions that hitherto (§ 5). A soultion is obtained for the busy period distribution in GI/G/1 (§ 7) which can be evaluated when either of the distributions concerned has a rational characteristic function. This solution contains some recent results of Conolly on the quene GI/En/1, as well as well-known results for M/G/1.

scholarly journals Single server queues with modified service mechanisms

1962 ◽  
Vol 2 (4) ◽  
pp. 499-507 ◽  
Author(s):  
G. F. Yeo
Keyword(s):  
Time Distribution ◽  
Busy Period ◽  
Queueing System ◽  
Probability Generating Function ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Period Distribution ◽  
Time Dependent Problem ◽  
Stationary Waiting Time ◽  
Special Case

SummaryThis paper considers a generalisation of the queueing system M/G/I, where customers arriving at empty and non-empty queues have different service time distributions. The characteristic function (c.f.) of the stationary waiting time distribution and the probability generating function (p.g.f.) of the queue size are obtained. The busy period distribution is found; the results are generalised to an Erlangian inter-arrival distribution; the time-dependent problem is considered, and finally a special case of server absenteeism is discussed.

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.

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.

CONTINUOUS-TIME FINANCE AND THE WAITING TIME DISTRIBUTION: MULTIPLE CHARACTERISTIC TIMES

2012 ◽  
Vol 26 (23) ◽  
pp. 1250151 ◽  
Author(s):  
KWOK SAU FA
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Waiting Time ◽  
Continuous Time ◽  
Probability Distribution Function ◽  
Time Distribution ◽  
Waiting Time Distribution ◽  
Exponential Functions ◽  
Multiple Characteristic ◽  
Sum Of Products

In this paper, we model the tick-by-tick dynamics of markets by using the continuous-time random walk (CTRW) model. We employ a sum of products of power law and stretched exponential functions for the waiting time probability distribution function; this function can fit well the waiting time distribution for BUND futures traded at LIFFE in 1997.

Another martingale bound on the waiting-time distribution in GI/G/1 queues

1979 ◽  
Vol 16 (2) ◽  
pp. 454-457 ◽  
Author(s):  
Harry H. Tan
Keyword(s):  
Waiting Time ◽  
Upper Bound ◽  
Time Distribution ◽  
Waiting Time Distribution ◽  
Martingale Approach ◽  
Stationary Waiting Time

A new upper bound on the stationary waiting-time distribution of a GI/G/1 queue is derived following Kingman's martingale approach. This bound is generally stronger than Kingman's upper bound and is sometimes stronger than an upper bound derived by Ross.

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