The effect of variability in the GI/G/s queue

1980 ◽  
Vol 17 (4) ◽  
pp. 1062-1071 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Waiting Time ◽  
Convex Functions ◽  
Waiting Times ◽  
Expected Value ◽  
Interarrival Time ◽  
Single Server ◽  
Server Systems ◽  
Clearing Time ◽  
Stationary Waiting Time ◽  
Time Required

In 1969 H. and D. Stoyan showed that the stationary waiting-time distribution in a GI/G/1 queue increases in the ordering determined by the expected value of all non-decreasing convex functions when the interarrival-time and service-time distributions become more variable, as expressed in the ordering determined by the expected value of all convex functions. Ross (1978) and Wolff (1977) showed by counterexample that this conclusion does not extend to all GI/G/s queues. Here it is shown that this conclusion does hold for all GI/G/s queues for several other measures of congestion which coincide with the waiting time in single-server systems. One such alternate measure of congestion is the clearing time, the time required after the arrival epoch of the nth customer for the system to serve all customers in the system at that time, excluding the nth customer. The stochastic comparisons also imply an ordering for the expected waiting times in M/G/s queues.

The effect of variability in the GI/G/s queue

1980 ◽  
Vol 17 (04) ◽  
pp. 1062-1071 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Waiting Time ◽  
Convex Functions ◽  
Waiting Times ◽  
Expected Value ◽  
Interarrival Time ◽  
Single Server ◽  
Server Systems ◽  
Clearing Time ◽  
Stationary Waiting Time ◽  
Time Required

In 1969 H. and D. Stoyan showed that the stationary waiting-time distribution in a GI/G/1 queue increases in the ordering determined by the expected value of all non-decreasing convex functions when the interarrival-time and service-time distributions become more variable, as expressed in the ordering determined by the expected value of all convex functions. Ross (1978) and Wolff (1977) showed by counterexample that this conclusion does not extend to all GI/G/s queues. Here it is shown that this conclusion does hold for all GI/G/s queues for several other measures of congestion which coincide with the waiting time in single-server systems. One such alternate measure of congestion is the clearing time, the time required after the arrival epoch of the nth customer for the system to serve all customers in the system at that time, excluding the nth customer. The stochastic comparisons also imply an ordering for the expected waiting times in M/G/s queues.

Some comparability results for waiting times in single- and many-server queues

1984 ◽  
Vol 21 (4) ◽  
pp. 887-900 ◽  
Author(s):  
D. J. Daley ◽  
T. Rolski
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Waiting Times ◽  
Queueing Systems ◽  
Random Variables ◽  
Interarrival Time ◽  
Sufficient Condition ◽  
Traffic Intensity ◽  
Many Server Queues ◽  
Stationary Waiting Time

It is shown that the stationary waiting time random variables W′, W″ of two M/G/l queueing systems for which the corresponding service time random variables satisfy E(S′−x)+ ≦ E(S″−x)+ (all x >0), are stochastically ordered as W′≦dW″. The weaker conclusion, that E(W′−x)+ ≦ E(W″−x)+ (all x > 0), is shown to hold in GI/M/k systems when the interarrival time random variables satisfy E(x−T′)+ ≦ E(x−T″)+ (all x). A sufficient condition for wk≡EW in GI/D/k to be monotonic in k for a sequence of k-server queues with the same relative traffic intensity is given. Evidence indicating or refuting possible strengthenings of some of the results is indicated.

scholarly journals SERVER WAITING TIMES IN INFINITE SUPPLY POLLING SYSTEMS WITH PREPARATION TIMES

2015 ◽  
Vol 30 (2) ◽  
pp. 153-184 ◽  
Author(s):  
Jan-Pieter L. Dorsman ◽  
Nir Perel ◽  
Maria Vlasiou
Keyword(s):  
Waiting Time ◽  
Queueing Networks ◽  
Waiting Times ◽  
Fixed Number ◽  
Polling Systems ◽  
Single Server ◽  
Dynamic Allocation ◽  
Mean Waiting Time ◽  
Preparation Phase ◽  
Stationary Waiting Time

We consider a system consisting of a single server serving a fixed number of stations. At each station, there is an infinite queue of customers that have to undergo a preparation phase before being served. This model is connected to layered queueing networks, to an extension of polling systems and surprisingly to random graphs. We are interested in the waiting time of the server. For the case where the server polls the stations cyclically, we give a sufficient condition for the existence of a limiting waiting-time distribution and we study the tail behavior of the stationary waiting time. Furthermore, assuming that preparation times are exponentially distributed, we describe in depth the resulting Markov chain. We also investigate a model variation where the server does not necessarily poll the stations in a cyclic order, but always serves the customer with the earliest completed preparation phase. We show that the mean waiting time under this dynamic allocation never exceeds that of the cyclic case, but that the waiting-time distributions corresponding to both cases are not necessarily stochastically ordered. Finally, we provide extensive numerical results investigating and comparing the effect of the system's parameters to the performance of the server for both models.

Some comparability results for waiting times in single- and many-server queues

1984 ◽  
Vol 21 (04) ◽  
pp. 887-900 ◽  
Author(s):  
D. J. Daley ◽  
T. Rolski
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Waiting Times ◽  
Queueing Systems ◽  
Random Variables ◽  
Interarrival Time ◽  
Sufficient Condition ◽  
Traffic Intensity ◽  
Many Server Queues ◽  
Stationary Waiting Time

It is shown that the stationary waiting time random variables W′, W″ of two M/G/l queueing systems for which the corresponding service time random variables satisfy E(S ′−x)+ ≦ E(S ″−x)+ (all x >0), are stochastically ordered as W ′≦d W ″. The weaker conclusion, that E(W ′−x)+ ≦ E(W ″−x)+ (all x > 0), is shown to hold in GI/M/k systems when the interarrival time random variables satisfy E(x−T ′)+ ≦ E(x−T ″)+ (all x). A sufficient condition for wk ≡EW in GI/D/k to be monotonic in k for a sequence of k-server queues with the same relative traffic intensity is given. Evidence indicating or refuting possible strengthenings of some of the results is indicated.

Convexity results for single-server queues and for multiserver queues with constant service times

1990 ◽  
Vol 27 (02) ◽  
pp. 465-468 ◽  
Author(s):  
Arie Harel
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Sojourn Time ◽  
Interarrival Time ◽  
Service Rate ◽  
Single Server ◽  
Multiserver Queues ◽  
Service Rates ◽  
Constant Service Times ◽  
Number Of Customers

We show that the waiting time in queue and the sojourn time of every customer in the G/G/1 and G/D/c queue are jointly convex in mean interarrival time and mean service time, and also jointly convex in mean interarrival time and service rate. Counterexamples show that this need not be the case, for the GI/GI/c queue or for the D/GI/c queue, for c ≧ 2. Also, we show that the average number of customers in the M/D/c queue is jointly convex in arrival and service rates. These results are surprising in light of the negative result for the GI/GI/2 queue (Weber (1983)).

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.

Convexity results for single-server queues and for multiserver queues with constant service times

1990 ◽  
Vol 27 (2) ◽  
pp. 465-468 ◽  
Author(s):  
Arie Harel
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Sojourn Time ◽  
Interarrival Time ◽  
Service Rate ◽  
Single Server ◽  
Multiserver Queues ◽  
Service Rates ◽  
Constant Service Times ◽  
Number Of Customers

We show that the waiting time in queue and the sojourn time of every customer in the G/G/1 and G/D/c queue are jointly convex in mean interarrival time and mean service time, and also jointly convex in mean interarrival time and service rate. Counterexamples show that this need not be the case, for the GI/GI/c queue or for the D/GI/c queue, for c ≧ 2. Also, we show that the average number of customers in the M/D/c queue is jointly convex in arrival and service rates.These results are surprising in light of the negative result for the GI/GI/2 queue (Weber (1983)).

Queues with semi-Markovian arrivals

1967 ◽  
Vol 4 (02) ◽  
pp. 365-379 ◽  
Author(s):  
Erhan Çinlar
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
Finite Number ◽  
Waiting Time ◽  
Busy Period ◽  
Queueing System ◽  
Interarrival Time ◽  
Single Server ◽  
Queue Size

A queueing system with a single server is considered. There are a finite number of types of customers, and the types of successive arrivals form a Markov chain. Further, the nth interarrival time has a distribution function which may depend on the types of the nth and the n–1th arrivals. The queue size, waiting time, and busy period processes are investigated. Both transient and limiting results are given.

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