An upper bound on the performance of queues with returning customers

1987 ◽  
Vol 24 (2) ◽  
pp. 466-475 ◽  
Author(s):  
Betsy S. Greenberg ◽  
Ronald W. Wolff
Keyword(s):  
Steady State ◽  
Upper Bound ◽  
System Performance ◽  
Finite Capacity ◽  
Multiple Channel ◽  
Poisson Arrivals ◽  
Time Averages ◽  
Steady State Probabilities

Multiple channel queues with Poisson arrivals, exponential service distributions, and finite capacity are studied. A customer who finds the system at capacity either leaves the system for ever or may return to try again after an exponentially distributed time. Steady state probabilities are approximated by assuming that the returning customers see time averages. The approximation is shown to result in an upper bound on system performance.

An upper bound on the performance of queues with returning customers

1987 ◽  
Vol 24 (02) ◽  
pp. 466-475 ◽  
Author(s):  
Betsy S. Greenberg ◽  
Ronald W. Wolff
Keyword(s):  
Steady State ◽  
Upper Bound ◽  
System Performance ◽  
Finite Capacity ◽  
Multiple Channel ◽  
Poisson Arrivals ◽  
Time Averages ◽  
Steady State Probabilities

Multiple channel queues with Poisson arrivals, exponential service distributions, and finite capacity are studied. A customer who finds the system at capacity either leaves the system for ever or may return to try again after an exponentially distributed time. Steady state probabilities are approximated by assuming that the returning customers see time averages. The approximation is shown to result in an upper bound on system performance.

M/G/1 queueing systems with returning customers

1989 ◽  
Vol 26 (01) ◽  
pp. 152-163 ◽  
Author(s):  
Betsy S. Greenberg
Keyword(s):  
Steady State ◽  
Laplace Transform ◽  
Single Channel ◽  
Queueing Systems ◽  
Service Distribution ◽  
Poisson Arrivals ◽  
General Service ◽  
Steady State Probabilities

Single-channel queues with Poisson arrivals, general service distributions, and no queue capacity are studied. A customer who finds the server busy either leaves the system for ever or may return to try again after an exponentially distributed time. Steady-state probabilities are approximated and bounded in two different ways. We characterize the service distribution by its Laplace transform, and use this characterization to determine the better method of approximation.

scholarly journals The slow server problem: a queue with stalling

1985 ◽  
Vol 22 (4) ◽  
pp. 879-892 ◽  
Author(s):  
Michael Rubinovitch
Keyword(s):  
Steady State ◽  
Stochastic Analysis ◽  
Server Problem ◽  
Poisson Arrivals ◽  
Steady State Probabilities

A queue with Poisson arrivals and two different exponential servers is considered. It is assumed that customers are allowed to stall, i.e., to wait for a busy fast server at times when the slow server is free. A stochastic analysis of the queue is given, steady-state probabilities are computed, and policies for overall optimization are characterized and computed. The issue of individual customer's optimization versus overall optimization is also discussed.

M/G/1 queueing systems with returning customers

1989 ◽  
Vol 26 (1) ◽  
pp. 152-163 ◽  
Author(s):  
Betsy S. Greenberg
Keyword(s):  
Steady State ◽  
Laplace Transform ◽  
Single Channel ◽  
Queueing Systems ◽  
Service Distribution ◽  
Poisson Arrivals ◽  
General Service ◽  
Steady State Probabilities

Single-channel queues with Poisson arrivals, general service distributions, and no queue capacity are studied. A customer who finds the server busy either leaves the system for ever or may return to try again after an exponentially distributed time. Steady-state probabilities are approximated and bounded in two different ways. We characterize the service distribution by its Laplace transform, and use this characterization to determine the better method of approximation.

Approximations for the steady-state probabilities in the M/G/c queue

1981 ◽  
Vol 13 (1) ◽  
pp. 186-206 ◽  
Author(s):  
H. C. Tijms ◽  
M. H. Van Hoorn ◽  
A. Federgruen
Keyword(s):  
Steady State ◽  
Queue Size ◽  
Numerical Experience ◽  
General Service Times ◽  
Server Queue ◽  
Poisson Arrivals ◽  
The Mean ◽  
Multi Server ◽  
General Service ◽  
Steady State Probabilities

For the multi-server queue with Poisson arrivals and general service times we present various approximations for the steady-state probabilities of the queue size. These approximations are computed from numerically stable recursion schemes which can be easily applied in practice. Numerical experience reveals that the approximations are very accurate with errors typically below 5%. For the delay probability the various approximations result either into the widely used Erlang delay probability or into a new approximation which improves in many cases the Erlang delay probability approximation. Also for the mean queue size we find a new approximation that turns out to be a good approximation for all values of the queueing parameters including the coefficient of variation of the service time.

Approximations for the steady-state probabilities in the M/G/c queue

1981 ◽  
Vol 13 (01) ◽  
pp. 186-206 ◽  
Author(s):  
H. C. Tijms ◽  
M. H. Van Hoorn ◽  
A. Federgruen
Keyword(s):  
Steady State ◽  
Queue Size ◽  
Numerical Experience ◽  
General Service Times ◽  
Server Queue ◽  
Poisson Arrivals ◽  
The Mean ◽  
Multi Server ◽  
General Service ◽  
Steady State Probabilities

For the multi-server queue with Poisson arrivals and general service times we present various approximations for the steady-state probabilities of the queue size. These approximations are computed from numerically stable recursion schemes which can be easily applied in practice. Numerical experience reveals that the approximations are very accurate with errors typically below 5%. For the delay probability the various approximations result either into the widely used Erlang delay probability or into a new approximation which improves in many cases the Erlang delay probability approximation. Also for the mean queue size we find a new approximation that turns out to be a good approximation for all values of the queueing parameters including the coefficient of variation of the service time.

scholarly journals Comparison of Inventory Systems with Service, Positive Lead-Time, Loss, and Retrial of Customers

2007 ◽  
Vol 2007 ◽  
pp. 1-23 ◽  
Author(s):  
A. Krishnamoorthy ◽  
K. P. Jose
Keyword(s):  
Steady State ◽  
System Performance ◽  
Lead Time ◽  
Inventory Systems ◽  
Inventory Level ◽  
Analytic Method ◽  
Finite Capacity ◽  
Inventory Models ◽  
Time Loss ◽  
Linear Rate

We analyze and compare three (s,S) inventory systems with positive service time and retrial of customers. In all of these systems, arrivals of customers form a Poisson process and service times are exponentially distributed. When the inventory level depletes to s due to services, an order of replenishment is placed. The lead-time follows an exponential distribution. In model I, an arriving customer, finding the inventory dry or server busy, proceeds to an orbit with probability γ and is lost forever with probability (1−γ). A retrial customer in the orbit, finding the inventory dry or server busy, returns to the orbit with probability δ and is lost forever with probability (1−δ). In addition to the description in model I, we provide a buffer of varying (finite) capacity equal to the current inventory level for model II and another having capacity equal to the maximum inventory level S for model III. In models II and III, an arriving customer, finding the buffer full, proceeds to an orbit with probability γ and is lost forever with probability (1−γ). A retrial customer in the orbit, finding the buffer full, returns to the orbit with probability δ and is lost forever with probability (1−δ). In all these models, the interretrial times are exponentially distributed with linear rate. Using matrix-analytic method, we study these inventory models. Some measures of the system performance in the steady state are derived. A suitable cost function is defined for all three cases and analyzed using graphical illustrations.

scholarly journals The slow server problem: a queue with stalling

1985 ◽  
Vol 22 (04) ◽  
pp. 879-892 ◽  
Author(s):  
Michael Rubinovitch
Keyword(s):  
Steady State ◽  
Stochastic Analysis ◽  
Server Problem ◽  
Poisson Arrivals ◽  
Steady State Probabilities

A queue with Poisson arrivals and two different exponential servers is considered. It is assumed that customers are allowed to stall, i.e., to wait for a busy fast server at times when the slow server is free. A stochastic analysis of the queue is given, steady-state probabilities are computed, and policies for overall optimization are characterized and computed. The issue of individual customer's optimization versus overall optimization is also discussed.

Tollbooth tandem queues with infinite homogeneous servers

2015 ◽  
Vol 52 (4) ◽  
pp. 941-961 ◽  
Author(s):  
Xiuli Chao ◽  
Qi-Ming He ◽  
Sheldon Ross
Keyword(s):  
Performance Measures ◽  
System Performance ◽  
Stochastic Ordering ◽  
Tandem Queues ◽  
Poisson Arrival ◽  
Arrival Processes ◽  
Poisson Arrivals ◽  
Number Of Customers ◽  
Steady State Solutions ◽  
Transient And Steady State

In this paper we analyze a tollbooth tandem queueing problem with an infinite number of servers. A customer starts service immediately upon arrival but cannot leave the system before all customers who arrived before him/her have left, i.e. customers depart the system in the same order as they arrive. Distributions of the total number of customers in the system, the number of departure-delayed customers in the system, and the number of customers in service at time t are obtained in closed form. Distributions of the sojourn times and departure delays of customers are also obtained explicitly. Both transient and steady state solutions are derived first for Poisson arrivals, and then extended to cases with batch Poisson and nonstationary Poisson arrival processes. Finally, we report several stochastic ordering results on how system performance measures are affected by arrival and service processes.

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