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

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.

Download Full-text

Approximating the stationary waiting time distribution function of GI/GI/1-queues with arithmetic interarrival time and service time distribution function

OR Spectrum ◽

10.1007/bf01720216 ◽

1982 ◽

Vol 4 (3) ◽

pp. 135-148 ◽

Cited By ~ 4

Author(s):

D. Wolf

Keyword(s):

Distribution Function ◽

Waiting Time ◽

Service Time ◽

Time Distribution ◽

Interarrival Time ◽

Service Time Distribution ◽

Waiting Time Distribution ◽

Stationary Waiting Time

Download Full-text

On Queues with Interarrival Times Proportional to Service Times

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800004198 ◽

1996 ◽

Vol 10 (1) ◽

pp. 87-107 ◽

Cited By ~ 7

Author(s):

Israel Cidon ◽

Roch Guréin ◽

Asad Khamisy ◽

Moshe Sidi

Keyword(s):

Waiting Time ◽

Service Time ◽

Queueing Systems ◽

Random Variable ◽

Interarrival Time ◽

Time Interval ◽

Equivalent System ◽

Stieltjes Transform ◽

Switched Networks ◽

Associated Solutions

We analyze a family of queueing systems where the interarrival time In+1 between customers n and n + 1 depends on the service time Bn of customer n. Specifically, we consider cases where the dependency between In+1 and Bn is a proportionality relation and Bn is an exponentially distributed random variable. Such dependencies arise in the context of packet-switched networks that use rate policing functions to regulate the amount of data that can arrive to a link within any given time interval. These controls result in significant dependencies between the amount of work brought in by customers/packets and the time between successive customers. The models developed in the paper and the associated solutions are, however, of independent interest and are potentially applicable to other environments.Several scenarios that consist of adding an independent random variable to the interarrival time, allowing the proportionality to be random and the combination of the two are considered. In all cases, we provide expressions for the Laplace-Stieltjes Transform of the waiting time of a customer in the system. Numerical results are provided and compared to those of an equivalent system without dependencies.

Download Full-text

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

Journal of Applied Probability ◽

10.1017/s0021900200097345 ◽

1980 ◽

Vol 17 (04) ◽

pp. 1062-1071 ◽

Cited By ~ 7

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.

Download Full-text

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

Journal of Applied Probability ◽

10.2307/3213215 ◽

1980 ◽

Vol 17 (4) ◽

pp. 1062-1071 ◽

Cited By ~ 28

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.

Download Full-text

Some general results for many server queues

Advances in Applied Probability ◽

10.1017/s0001867800039008 ◽

1973 ◽

Vol 5 (01) ◽

pp. 153-169 ◽

Cited By ~ 8

Author(s):

J. H. A. De Smit

Keyword(s):

Integral Equations ◽

Waiting Time ◽

Queue Length ◽

Joint Distribution ◽

Waiting Times ◽

Virtual Waiting Time ◽

Server Queue ◽

Many Server Queues ◽

The Many ◽

Busy Cycle

Pollaczek's theory for the many server queue is generalized and extended. Pollaczek (1961) found the distribution of the actual waiting times in the model G/G/s as a solution of a set of integral equations. We give a somewhat more general set of integral equations from which the joint distribution of the actual waiting time and some other random variables may be found. With this joint distribution we can obtain distributions of a number of characteristic quantities, such as the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. For a wide class of many server queues the formal expressions may lead to explicit results.

Download Full-text

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

Journal of Applied Probability ◽

10.1017/s0021900200038948 ◽

1990 ◽

Vol 27 (02) ◽

pp. 465-468 ◽

Cited By ~ 1

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)).

Download Full-text

On a property of a refusals stream

Journal of Applied Probability ◽

10.1017/s0021900200101469 ◽

1997 ◽

Vol 34 (03) ◽

pp. 800-805 ◽

Cited By ~ 5

Author(s):

Vyacheslav M. Abramov

Keyword(s):

Waiting Time ◽

Service Time ◽

Busy Period ◽

Elementary Proof ◽

Queueing System ◽

Queueing Systems ◽

Single Server ◽

Wide Family

This paper consists of two parts. The first part provides a more elementary proof of the asymptotic theorem of the refusals stream for an M/GI/1/n queueing system discussed in Abramov (1991a). The central property of the refusals stream discussed in the second part of this paper is that, if the expectations of interarrival and service time of an M/GI/1/n queueing system are equal to each other, then the expectation of the number of refusals during a busy period is equal to 1. This property is extended for a wide family of single-server queueing systems with refusals including, for example, queueing systems with bounded waiting time.

Download Full-text

Sequential Appointment Scheduling Considering Walk-In Patients

Mathematical Problems in Engineering ◽

10.1155/2014/564832 ◽

2014 ◽

Vol 2014 ◽

pp. 1-12 ◽

Cited By ~ 4

Author(s):

Chongjun Yan ◽

Jiafu Tang ◽

Bowen Jiang

Keyword(s):

Objective Function ◽

Waiting Time ◽

Service Time ◽

Appointment Scheduling ◽

Optimal Scheduling ◽

Sufficient Condition ◽

Profit Function ◽

Number Of Patients ◽

Service Efficiency ◽

Appointment Time

This paper develops a sequential appointment algorithm considering walk-in patients. In practice, the scheduler assigns an appointment time for each call-in patient before the call ends, and the appointment time cannot be changed once it is set. Each patient has a certain probability of being a no-show patient on the day of appointment. The objective is to determine the optimal booking number of patients and the optimal scheduling time for each patient to maximize the revenue of all the arriving patients minus the expenses of waiting time and overtime. Based on the assumption that the service time is exponentially distributed, this paper proves that the objective function is convex. A sufficient condition under which the profit function is unimodal is provided. The numerical results indicate that the proposed algorithm outperforms all the commonly used heuristics, lowering the instances of no-shows, and walk-in patients can improve the service efficiency and bring more profits to the clinic. It is also noted that the potential appointment is an effective alternative to mitigate no-show phenomenon.

Download Full-text

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

Journal of Applied Probability ◽

10.2307/3214668 ◽

1990 ◽

Vol 27 (2) ◽

pp. 465-468 ◽

Cited By ~ 5

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)).

Download Full-text