Stationary Poisson departure processes from non-stationary queues

1986 ◽  
Vol 23 (1) ◽  
pp. 256-260 ◽  
Author(s):  
Robert D. Foley
Keyword(s):  
Poisson Process ◽  
Service Time ◽  
Queueing Systems ◽  
Arrival Rate ◽  
Distribution Functions ◽  
Arrival Process ◽  
Service Time Distribution ◽  
Departure Process ◽  
Infinite Server ◽  
Departure Processes

We present some non-stationary infinite-server queueing systems with stationary Poisson departure processes. In Foley (1982), it was shown that the departure process from the Mt/Gt/∞ queue was a Poisson process, possibly non-stationary. The Mt/Gt/∞ queue is an infinite-server queue with a stationary or non-stationary Poisson arrival process and a general server in which the service time of a customer may depend upon the customer's arrival time. Mirasol (1963) pointed out that the departure process from the M/G/∞ queue is a stationary Poisson process. The question arose whether there are any other Mt/Gt/∞ queueing systems with stationary Poisson departure processes. For example, if the arrival rate is periodic, is it possible to select the service-time distribution functions to fluctuate in order to compensate for the fluctuations of the arrival rate? In this situation and in more general situations, it is possible to select the server such that the system yields a stationary Poisson departure process.

Stationary Poisson departure processes from non-stationary queues

1986 ◽  
Vol 23 (01) ◽  
pp. 256-260 ◽  
Author(s):  
Robert D. Foley
Keyword(s):  
Poisson Process ◽  
Service Time ◽  
Queueing Systems ◽  
Arrival Rate ◽  
Distribution Functions ◽  
Arrival Process ◽  
Service Time Distribution ◽  
Departure Process ◽  
Infinite Server ◽  
Departure Processes

We present some non-stationary infinite-server queueing systems with stationary Poisson departure processes. In Foley (1982), it was shown that the departure process from the Mt/Gt/∞ queue was a Poisson process, possibly non-stationary. The Mt/Gt /∞ queue is an infinite-server queue with a stationary or non-stationary Poisson arrival process and a general server in which the service time of a customer may depend upon the customer's arrival time. Mirasol (1963) pointed out that the departure process from the M/G/∞ queue is a stationary Poisson process. The question arose whether there are any other Mt/Gt/∞ queueing systems with stationary Poisson departure processes. For example, if the arrival rate is periodic, is it possible to select the service-time distribution functions to fluctuate in order to compensate for the fluctuations of the arrival rate? In this situation and in more general situations, it is possible to select the server such that the system yields a stationary Poisson departure process.

On queueing systems with renewal departure processes

1983 ◽  
Vol 15 (3) ◽  
pp. 657-673 ◽  
Author(s):  
Mark Berman ◽  
Mark Westcott
Keyword(s):  
Queueing Systems ◽  
Sufficient Conditions ◽  
Arrival Process ◽  
Necessary Condition ◽  
Size Distributions ◽  
Departure Process ◽  
Arrival Processes ◽  
Departure Processes ◽  
Interval Distribution

It is proved that, for a large class of stable stationary queueing systems with renewal arrival processes and without losses, a necessary condition for the departure process also to be a renewal process is that its interval distribution be the same as that of the arrival process. This result is then applied to the classical GI/G/s queueing systems. In particular, alternative proofs of known characterizations of the M/G/1 and GI/M/1 systems are given, as well as a characterization of the GI/G/∞ system. In the course of the proofs, sufficient conditions for the existence of all the moments of the stationary queue-size distributions of both the GI/G/1 and GI/G/∞ systems are derived.

On queueing systems with renewal departure processes

1983 ◽  
Vol 15 (03) ◽  
pp. 657-673 ◽  
Author(s):  
Mark Berman ◽  
Mark Westcott
Keyword(s):  
Queueing Systems ◽  
Sufficient Conditions ◽  
Arrival Process ◽  
Necessary Condition ◽  
Size Distributions ◽  
Departure Process ◽  
Arrival Processes ◽  
Departure Processes ◽  
Interval Distribution

It is proved that, for a large class of stable stationary queueing systems with renewal arrival processes and without losses, a necessary condition for the departure process also to be a renewal process is that its interval distribution be the same as that of the arrival process. This result is then applied to the classicalGI/G/squeueing systems. In particular, alternative proofs of known characterizations of theM/G/1 andGI/M/1 systems are given, as well as a characterization of theGI/G/∞ system. In the course of the proofs, sufficient conditions for the existence of all the moments of the stationary queue-size distributions of both theGI/G/1 andGI/G/∞ systems are derived.

Hitting time in an M/G/1 queue

1999 ◽  
Vol 36 (03) ◽  
pp. 934-940 ◽  
Author(s):  
Sheldon M. Ross ◽  
Sridhar Seshadri
Keyword(s):  
Service Time ◽  
Time Distribution ◽  
Arrival Rate ◽  
Hitting Time ◽  
Service Time Distribution ◽  
Expected Time ◽  
Efficient Simulation ◽  
Simulation Procedure ◽  
Poisson Arrival ◽  
Stationary Delay

We study the expected time for the work in an M/G/1 system to exceed the level x, given that it started out initially empty, and show that it can be expressed solely in terms of the Poisson arrival rate, the service time distribution and the stationary delay distribution of the M/G/1 system. We use this result to construct an efficient simulation procedure.

A stationary Poisson departure process from a minimally delayed infinite server queue with non-stationary Poisson arrivals

1997 ◽  
Vol 34 (3) ◽  
pp. 767-772 ◽  
Author(s):  
John A. Barnes ◽  
Richard Meili
Keyword(s):  
Queueing System ◽  
Mean Shift ◽  
Intensity Function ◽  
Arrival Process ◽  
Service Time Distribution ◽  
Departure Process ◽  
Periodic Intensity ◽  
Server Queue ◽  
Poisson Arrivals ◽  
The Mean

The points of a non-stationary Poisson process with periodic intensity are independently shifted forward in time in such a way that the transformed process is stationary Poisson. The mean shift is shown to be minimal. The approach used is to consider an Mt/Gt/∞ queueing system where the arrival process is a non-stationary Poisson with periodic intensity function. A minimal service time distribution is constructed that yields a stationary Poisson departure process.

On extremal service disciplines in single-stage queueing systems

1990 ◽  
Vol 27 (02) ◽  
pp. 409-416 ◽  
Author(s):  
Rhonda Righter ◽  
J. George Shanthikumar ◽  
Genji Yamazaki
Keyword(s):  
Service Time ◽  
Sojourn Time ◽  
Queueing System ◽  
Residual Life ◽  
Queueing Systems ◽  
Mean Residual Life ◽  
Service Discipline ◽  
Service Time Distribution ◽  
Service Disciplines ◽  
Number Of Customers

It is shown that among all work-conserving service disciplines that are independent of the future history, the first-come-first-served (FCFS) service discipline minimizes [maximizes] the average sojourn time in a G/GI/1 queueing system with new better [worse] than used in expectation (NBUE[NWUE]) service time distribution. We prove this result using a new basic identity of G/GI/1 queues that may be of independent interest. Using a relationship between the workload and the number of customers in the system with different lengths of attained service it is shown that the average sojourn time is minimized [maximized] by the least-attained-service time (LAST) service discipline when the service time has the decreasing [increasing] mean residual life (DMRL[IMRL]) property.

The convexity of a general performance measure for multiserver queues

1987 ◽  
Vol 24 (03) ◽  
pp. 725-736 ◽  
Author(s):  
Arie Harel ◽  
Paul Zipkin
Keyword(s):  
Queueing Systems ◽  
Arrival Rate ◽  
Performance Measure ◽  
Service Rate ◽  
Service Time Distribution ◽  
Multiserver Queues ◽  
General Performance ◽  
Production Models ◽  
Convexity Properties ◽  
Special Case

This paper examines a general performance measure for queueing systems. This criterion reflects both the mean and the variance of sojourn times; the standard deviation is a special case. The measure plays a key role in certain production models, and it should be useful in a variety of other applications. We focus here on convexity properties of an approximation of the measure for the M/G/c queue. For c ≧ 2 we show that this quantity is convex in the arrival rate. Assuming the service rate acts as a scale factor in the service-time distribution, the measure is convex in the service rate also.

scholarly journals s,S Inventory with Positive Service Time and Retrial of Demands: An Approach through Multiserver Queues

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Anoop N. Nair ◽  
M. J. Jacob
Keyword(s):  
Steady State ◽  
Poisson Process ◽  
Service Time ◽  
Time Distribution ◽  
Joint Distribution ◽  
Service Time Distribution ◽  
Exponential Rate ◽  
Numerical Illustration ◽  
Multiserver Queues ◽  
Geometric Methods

We analyze an s,S inventory with positive service time and retrial of demands by considering the inventory as servers of a multiserver queuing system. Demands arrive according to a Poisson process and service time distribution is exponential. On each service completion, the number of demands in the system as well as the number of inventories (servers) is reduced by one. When all servers are busy, new arrivals join an orbit from which they try to access the service at an exponential rate. Using matrix geometric methods the steady state joint distribution of the demands and inventory has been analyzed and a numerical illustration is given.

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