scholarly journals The Waiting-Time Distribution for the GI/G/1 Queue under the D-Policy

1992 ◽  
Vol 6 (3) ◽  
pp. 287-308 ◽  
Author(s):  
Jingwen Li ◽  
Shun-Chen Niu
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Busy Period ◽  
Waiting Times ◽  
Optimization Models ◽  
Arrival Process ◽  
Waiting Time Distribution ◽  
Service Times

We study a generalization of the GI/G/l queue in which the server is turned off at the end of each busy period and is reactivated only when the sum of the service times of all waiting customers exceeds a given threshold of size D. Using the concept of a “randomly selected” arriving customer, we obtain as our main result a relation that expresses the waiting-time distribution of customers in this model in terms of characteristics associated with a corresponding standard GI/G/1 queue, obtained by setting D = 0. If either the arrival process is Poisson or the service times are exponentially distributed, then this representation of the waiting-time distribution can be specialized to yield explicit, transform-free formulas; we also derive, in both of these cases, the expected customer waiting times. Our results are potentially useful, for example, for studying optimization models in which the threshold D can be controlled.

Convex ordering of the attained waiting times in single-server queues and related problems

1991 ◽  
Vol 28 (02) ◽  
pp. 433-445 ◽  
Author(s):  
Masakiyo Miyazawa ◽  
Genji Yamazaki
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Waiting Times ◽  
Service Discipline ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Convex Order ◽  
Convex Ordering ◽  
Single Server Queues ◽  
Service Disciplines

The attained waiting time of customers in service of the G/G/1 queue is compared for various work-conserving service disciplines. It is proved that the attained waiting time distribution is minimized (maximized) in convex order when the discipline is FCFS (PR-LCFS). We apply the result to characterize finiteness of moments of the attained waiting time in the GI/GI/1 queue with an arbitrary work-conserving service discipline. In this discussion, some interesting relationships are obtained for a PR-LCFS queue.

scholarly journals The Workload in the M/G/1 Queue with Work Removal

1996 ◽  
Vol 10 (2) ◽  
pp. 261-277 ◽  
Author(s):  
Richard J. Boucherie ◽  
Onno J. Boxma
Keyword(s):  
Poisson Process ◽  
Waiting Time ◽  
Time Distribution ◽  
Arrival Process ◽  
Waiting Time Distribution ◽  
Negative Customers ◽  
Workload Distribution

We consider an M/G/1 queue with the special feature of additional negative customers, who arrive according to a Poisson process. Negative customers require no service, but at their arrival a stochastic amount of work is instantaneously removed from the system. We show that the workload distribution in this M/G/1 queue with negative customers equals the waiting time distribution in a GI/G/1 queue with ordinary customers only; the effect of the negative customers is incorporated in the new arrival process.

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.

On queues in which customers are served in random order

1962 ◽  
Vol 58 (1) ◽  
pp. 79-91 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Steady State ◽  
Exponential Decay ◽  
Waiting Time ◽  
Time Distribution ◽  
Busy Period ◽  
Asymptotic Form ◽  
Heavy Traffic ◽  
Formal Solution ◽  
Random Order ◽  
Waiting Time Distribution

ABSTRACTThe queue M |G| l is considered in the case in which customers are served in random order. A formal solution is obtained for the waiting time distribution in the steady state, and is used to consider the exponential decay of the distribution. The moments of the waiting time are examined, and the asymptotic form of the distribution in heavy traffic is found. Finally, the problem is related to those of the busy period and the approach to the steady state.

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.

scholarly journals M/G/1/K system with push-out scheme under vacation policy

1996 ◽  
Vol 9 (2) ◽  
pp. 143-157 ◽  
Author(s):  
Shoji Kasahara ◽  
Hideaki Takagi ◽  
Yutaka Takahashi ◽  
Toshiharu Hasegawa
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Waiting Times ◽  
Numerical Results ◽  
The Other ◽  
Waiting Time Distribution ◽  
Stieltjes Transform ◽  
Multiple Vacations ◽  
Push Out

We consider an M/G/1/K system with push-out scheme and multiple vacations. This model is particularly important in situations where it is essential to provide short waiting times to messages which are selected for service. We analyze the behavior of two types of messages: one that succeeds in transmission and the other that fails. We derive the Laplace-Stieltjes transform of the waiting time distribution for the message which is eventually served. Finally, we show some numerical results including the comparisons between the push-out and the ordinary blocking models.

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