Clustering of events in a stochastic process

1981 ◽  
Vol 18 (1) ◽  
pp. 268-275 ◽  
Author(s):  
Joseph Glaz
Keyword(s):  
Stochastic Process ◽  
Waiting Time ◽  
Service Time ◽  
Exact Distribution ◽  
Fixed Interval ◽  
Server Queue ◽  
Clustering Of Events ◽  
Poisson Arrivals ◽  
Expected Waiting Time ◽  
Expected Duration

In this paper we derive bounds for the expected waiting time of clustering of at least n events of a stochastic process within a fixed interval of length p. Using this approach of clustering, we derive bounds for the expected duration of the period of time that at least n servers are busy in an ∞-server queue with constant service time. For the case of Poisson arrivals we derive the exact distribution of the duration of that period.

Clustering of events in a stochastic process

1981 ◽  
Vol 18 (01) ◽  
pp. 268-275 ◽  
Author(s):  
Joseph Glaz
Keyword(s):  
Stochastic Process ◽  
Waiting Time ◽  
Service Time ◽  
Exact Distribution ◽  
Fixed Interval ◽  
Server Queue ◽  
Clustering Of Events ◽  
Poisson Arrivals ◽  
Expected Waiting Time ◽  
Expected Duration

In this paper we derive bounds for the expected waiting time of clustering of at least n events of a stochastic process within a fixed interval of length p. Using this approach of clustering, we derive bounds for the expected duration of the period of time that at least n servers are busy in an ∞-server queue with constant service time. For the case of Poisson arrivals we derive the exact distribution of the duration of that period.

scholarly journals An M/G/1 queue with multiple types of feedback and gated vacations

1997 ◽  
Vol 34 (03) ◽  
pp. 773-784 ◽  
Author(s):  
Onno J. Boxma ◽  
Uri Yechiali
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Queue Length ◽  
Time Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Single Server Queue ◽  
Server Queue ◽  
Poisson Arrivals

This paper considers a single-server queue with Poisson arrivals and multiple customer feedbacks. If the first service attempt of a newly arriving customer is not successful, he returns to the end of the queue for another service attempt, with a different service time distribution. He keeps trying in this manner (as an ‘old' customer) until his service is successful. The server operates according to the ‘gated vacation' strategy; when it returns from a vacation to find K (new and old) customers, it renders a single service attempt to each of them and takes another vacation, etc. We study the joint queue length process of new and old customers, as well as the waiting time distribution of customers. Some extensions are also discussed.

scholarly journals An M/G/1 queue with multiple types of feedback and gated vacations

1997 ◽  
Vol 34 (3) ◽  
pp. 773-784 ◽  
Author(s):  
Onno J. Boxma ◽  
Uri Yechiali
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Queue Length ◽  
Time Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Single Server Queue ◽  
Server Queue ◽  
Poisson Arrivals

This paper considers a single-server queue with Poisson arrivals and multiple customer feedbacks. If the first service attempt of a newly arriving customer is not successful, he returns to the end of the queue for another service attempt, with a different service time distribution. He keeps trying in this manner (as an ‘old' customer) until his service is successful. The server operates according to the ‘gated vacation' strategy; when it returns from a vacation to find K (new and old) customers, it renders a single service attempt to each of them and takes another vacation, etc. We study the joint queue length process of new and old customers, as well as the waiting time distribution of customers. Some extensions are also discussed.

On the many server queue with exponential service times

1973 ◽  
Vol 5 (1) ◽  
pp. 170-182 ◽  
Author(s):  
J. H. A. De Smit
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Busy Period ◽  
Virtual Waiting Time ◽  
Server Queue ◽  
Poisson Arrivals ◽  
Service Times ◽  
The Many ◽  
Special Case ◽  
Busy Cycle

The general theory for the many server queue due to Pollaczek (1961) and generalized by the author (de Smit (1973)) is applied to the system with exponential service times. In this way many explicit results are obtained for the distributions of characteristic quantities, such as the actual waiting time, the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. Most of these results are new, even for the special case of Poisson arrivals.

On the many server queue with exponential service times

1973 ◽  
Vol 5 (01) ◽  
pp. 170-182 ◽  
Author(s):  
J. H. A. De Smit
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Busy Period ◽  
Virtual Waiting Time ◽  
Server Queue ◽  
Poisson Arrivals ◽  
Service Times ◽  
The Many ◽  
Special Case ◽  
Busy Cycle

The general theory for the many server queue due to Pollaczek (1961) and generalized by the author (de Smit (1973)) is applied to the system with exponential service times. In this way many explicit results are obtained for the distributions of characteristic quantities, such as the actual waiting time, the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. Most of these results are new, even for the special case of Poisson arrivals.

On the single-server queue with non-homogeneous Poisson input and general service time

1964 ◽  
Vol 1 (2) ◽  
pp. 369-384 ◽  
Author(s):  
A. M. Hasofer
Keyword(s):  
Periodic Function ◽  
Waiting Time ◽  
Service Time ◽  
Single Server ◽  
Single Server Queue ◽  
Poisson Input ◽  
Server Queue ◽  
Asymptotically Periodic ◽  
Image Position ◽  
General Service

In this paper, a single-server queue with non-homogeneous Poisson input and general service time is considered. Particular attention is given to the case where the parameter of the Poisson input λ(t) is a periodic function of the time. The approach is an extension of the work of Takács and Reich . The main result of the investigation is that under certain conditions on the distribution of the service time, the form of the function λ(t) and the distribution of the waiting time at t = 0, the probability of a server being idle P0 and the Laplace transform Ω of the waiting time are both asymptotically periodic in t. Putting where b(t) is a periodic function of time, it is shown that both Po and Ω can be expanded in a power series in z, and a method for calculating explicitly the asymptotic values of the leading terms is obtained.In many practical queueing problems, it is expected that the probability of arrivals will vary periodically. For example, in restaurants or at servicestations arrivals are more probable at rush hours than at slack periods, and rush hours are repeated day after day

On the single-server queue with non-homogeneous Poisson input and general service time

1964 ◽  
Vol 1 (02) ◽  
pp. 369-384 ◽  
Author(s):  
A. M. Hasofer
Keyword(s):  
Periodic Function ◽  
Waiting Time ◽  
Service Time ◽  
Single Server ◽  
Single Server Queue ◽  
Poisson Input ◽  
Server Queue ◽  
Asymptotically Periodic ◽  
Image Position ◽  
General Service

In this paper, a single-server queue with non-homogeneous Poisson input and general service time is considered. Particular attention is given to the case where the parameter of the Poisson input λ(t) is a periodic function of the time. The approach is an extension of the work of Takács and Reich . The main result of the investigation is that under certain conditions on the distribution of the service time, the form of the function λ(t) and the distribution of the waiting time at t = 0, the probability of a server being idle P 0 and the Laplace transform Ω of the waiting time are both asymptotically periodic in t. Putting where b(t) is a periodic function of time, it is shown that both P o and Ω can be expanded in a power series in z, and a method for calculating explicitly the asymptotic values of the leading terms is obtained. In many practical queueing problems, it is expected that the probability of arrivals will vary periodically. For example, in restaurants or at servicestations arrivals are more probable at rush hours than at slack periods, and rush hours are repeated day after day

scholarly journals MAD Dispersion Measure Makes Extremal Queue Analysis Simple

2022 ◽  
Author(s):  
Wouter van Eekelen ◽  
Dick den Hertog ◽  
Johan S.H. van Leeuwaarden
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Queueing Theory ◽  
Upper Bounds ◽  
Dispersion Measure ◽  
Mean Absolute Deviation ◽  
Absolute Deviation ◽  
Worst Case ◽  
Expected Waiting Time ◽  
Computational Intractability

A notorious problem in queueing theory is to compute the worst possible performance of the GI/G/1 queue under mean-dispersion constraints for the interarrival- and service-time distributions. We address this extremal queue problem by measuring dispersion in terms of mean absolute deviation (MAD) instead of the more conventional variance, making available methods for distribution-free analysis. Combined with random walk theory, we obtain explicit expressions for the extremal interarrival- and service-time distributions and, hence, the best possible upper bounds for all moments of the waiting time. We also obtain tight lower bounds that, together with the upper bounds, provide robust performance intervals. We show that all bounds are computationally tractable and remain sharp also when the mean and MAD are not known precisely but are estimated based on available data instead. Summary of Contribution: Queueing theory is a classic OR topic with a central role for the GI/G/1 queue. Although this queueing system is conceptually simple, it is notoriously hard to determine the worst-case expected waiting time when only knowing the first two moments of the interarrival- and service-time distributions. In this setting, the exact form of the extremal distribution can only be determined numerically as the solution to a nonconvex nonlinear optimization problem. Our paper demonstrates that using mean absolute deviation (MAD) instead of variance alleviates the computational intractability of the extremal GI/G/1 queue problem, enabling us to state the worst-case distributions explicitly.

Some general results for many server queues

1973 ◽  
Vol 5 (01) ◽  
pp. 153-169 ◽  
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.

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