Queues with negative arrivals

1991 ◽  
Vol 28 (1) ◽  
pp. 245-250 ◽  
Author(s):  
Erol Gelenbe ◽  
Peter Glynn ◽  
Karl Sigman
Keyword(s):  
Stability Conditions ◽  
Queueing Models ◽  
Single Server ◽  
New Models ◽  
Service Rates ◽  
Service Times ◽  
The Stability ◽  
Negative Arrivals ◽  
The Way

We study single-server queueing models where in addition to regular arriving customers, there are negative arrivals. A negative arrival has the effect of removing a customer from the queue. The way in which this removal is specified gives rise to several different models. Unlike the standard FIFOGI/GI/1 model, the stability conditions for these new models may depend upon more than just the arrival and service rates; the entire distributions of interarrival and service times may be involved.

Queues with negative arrivals

1991 ◽  
Vol 28 (01) ◽  
pp. 245-250 ◽  
Author(s):  
Erol Gelenbe ◽  
Peter Glynn ◽  
Karl Sigman
Keyword(s):  
Stability Conditions ◽  
Queueing Models ◽  
Single Server ◽  
New Models ◽  
Service Rates ◽  
Service Times ◽  
The Stability ◽  
Negative Arrivals ◽  
The Way

We study single-server queueing models where in addition to regular arriving customers, there are negative arrivals. A negative arrival has the effect of removing a customer from the queue. The way in which this removal is specified gives rise to several different models. Unlike the standard FIFO GI/GI/1 model, the stability conditions for these new models may depend upon more than just the arrival and service rates; the entire distributions of interarrival and service times may be involved.

The M/G/1 queue with negative customers

1996 ◽  
Vol 28 (02) ◽  
pp. 540-566 ◽  
Author(s):  
Peter G. Harrison ◽  
Edwige Pitel
Keyword(s):  
Iterative Solution ◽  
Single Server ◽  
Negative Customers ◽  
Poisson Arrival ◽  
First Case ◽  
Full Study ◽  
Service Times ◽  
The Stability ◽  
The One ◽  
Iterative Solution Method

We derive expressions for the generating function of the equilibrium queue length probability distribution in a single server queue with general service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. For the case of first come first served queueing discipline for the positive customers, we compare the killing strategies in which either the last customer in the queue or the one in service is removed by a negative customer. We then consider preemptive-restart with resampling last come first served queueing discipline for the positive customers, combined with the elimination of the customer in service by a negative customer—the case of elimination of the last customer yields an analysis similar to first come first served discipline for positive customers. The results show different generating functions in contrast to the case where service times are exponentially distributed. This is also reflected in the stability conditions. Incidently, this leads to a full study of the preemptive-restart with resampling last come first served case without negative customers. Finally, approaches to solving the Fredholm integral equation of the first kind which arises, for instance, in the first case are considered as well as an alternative iterative solution method.

scholarly journals On the stability of polling models with multiple servers

1998 ◽  
Vol 35 (04) ◽  
pp. 925-935 ◽  
Author(s):  
D. Down
Keyword(s):  
Stability Conditions ◽  
Single Server ◽  
Fluid Limit ◽  
Polling Models ◽  
Multiple Servers ◽  
The Stability ◽  
Limit Models

The stability of polling models is examined using associated fluid limit models. Examples are presented which generalize existing results in the literature or provide new stability conditions while in both cases providing simple and intuitive proofs of stability. The analysis is performed for both general single server models and specific multiple server models.

The matrix M/M/∞ system: retrial models and Markov Modulated sources

1993 ◽  
Vol 25 (2) ◽  
pp. 453-471 ◽  
Author(s):  
J. Keilson ◽  
L. D. Servi
Keyword(s):  
Numerical Evaluation ◽  
Stability Conditions ◽  
Kummer Functions ◽  
The Matrix ◽  
Service Rates ◽  
System Properties ◽  
Markov Modulated ◽  
The Stability ◽  
Analytical Expressions

The matrix-geometric work of Neuts could be viewed as a matrix variant of M/M/1. A 2 × 2 matrix counterpart of Neuts for M/M/∞ is introduced, the stability conditions are identified, and the ergodic solution is solved analytically in terms of the ten parameters that define it. For several cases of interest, system properties can be found from simple analytical expressions or after easy numerical evaluation of Kummer functions. When the matrix of service rates is singular, a qualitatively different solution is derived. Applications to telecommunications include some retrial models and an M/M/∞ queue with Markov-modulated input.

Stochastic comparisons for fork-join queues with exponential processing times

1997 ◽  
Vol 34 (2) ◽  
pp. 487-497 ◽  
Author(s):  
Esther Frostig ◽  
Tapani Lehtonen
Keyword(s):  
Random Variables ◽  
Single Server ◽  
Departure Process ◽  
Stochastic Comparisons ◽  
Processing Times ◽  
Service Rates ◽  
Service Times ◽  
Stochastic Majorization

Consider a fork-join queue, where each job upon arrival splits into k tasks and each joins a separate queue that is attended by a single server. Service times are independent, exponentially distributed random variables. Server i works at rate , where μ is constant. We prove that the departure process becomes stochastically faster as the service rates become more homogeneous in the sense of stochastic majorization. Consequently, when all k servers work with equal rates the departure process is stochastically maximized.

The M/G/1 queue with negative customers

1996 ◽  
Vol 28 (2) ◽  
pp. 540-566 ◽  
Author(s):  
Peter G. Harrison ◽  
Edwige Pitel
Keyword(s):  
Iterative Solution ◽  
Single Server ◽  
Negative Customers ◽  
Poisson Arrival ◽  
First Case ◽  
Full Study ◽  
Service Times ◽  
The Stability ◽  
The One ◽  
Iterative Solution Method

We derive expressions for the generating function of the equilibrium queue length probability distribution in a single server queue with general service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. For the case of first come first served queueing discipline for the positive customers, we compare the killing strategies in which either the last customer in the queue or the one in service is removed by a negative customer. We then consider preemptive-restart with resampling last come first served queueing discipline for the positive customers, combined with the elimination of the customer in service by a negative customer—the case of elimination of the last customer yields an analysis similar to first come first served discipline for positive customers. The results show different generating functions in contrast to the case where service times are exponentially distributed. This is also reflected in the stability conditions. Incidently, this leads to a full study of the preemptive-restart with resampling last come first served case without negative customers. Finally, approaches to solving the Fredholm integral equation of the first kind which arises, for instance, in the first case are considered as well as an alternative iterative solution method.

scholarly journals Sample-path stability conditions for multiserver input-output processes

1994 ◽  
Vol 7 (3) ◽  
pp. 437-456 ◽  
Author(s):  
Muhammad El-Taha ◽  
Shaler Stidham
Keyword(s):  
A Priori ◽  
Sample Path ◽  
Stability Conditions ◽  
Single Server ◽  
Input Output ◽  
State Spaces ◽  
Multiserver Queues ◽  
Path Stability ◽  
The Stability ◽  
Little's Formula

We extend our studies of sample-path stability to multiserver input-output processes with conditional output rates that may depend on the state of the system and other auxiliary processes. Our results include processes with countable as well as uncountable state spaces. We establish rate stability conditions for busy period durations as well as the input during busy periods. In addition, stability conditions for multiserver queues with possibly heterogeneous servers are given for the workload, attained service, and queue length processes. The stability conditions can be checked from parameters of primary processes, and thus can be verified a priori. Under the rate stability conditions, we provide stable versions of Little's formula for single server as well as multiserver queues. Our approach leads to extensions of previously known results. Since our results are valid pathwise, non-stationary as well as stationary processes are covered.

scholarly journals On the stability of polling models with multiple servers

1998 ◽  
Vol 35 (4) ◽  
pp. 925-935 ◽  
Author(s):  
D. Down
Keyword(s):  
Stability Conditions ◽  
Single Server ◽  
Fluid Limit ◽  
Polling Models ◽  
Multiple Servers ◽  
The Stability ◽  
Limit Models

The stability of polling models is examined using associated fluid limit models. Examples are presented which generalize existing results in the literature or provide new stability conditions while in both cases providing simple and intuitive proofs of stability. The analysis is performed for both general single server models and specific multiple server models.

Stochastic comparisons for fork-join queues with exponential processing times

1997 ◽  
Vol 34 (02) ◽  
pp. 487-497 ◽  
Author(s):  
Esther Frostig ◽  
Tapani Lehtonen
Keyword(s):  
Random Variables ◽  
Single Server ◽  
Departure Process ◽  
Stochastic Comparisons ◽  
Processing Times ◽  
Service Rates ◽  
Service Times ◽  
Stochastic Majorization

Consider a fork-join queue, where each job upon arrival splits into k tasks and each joins a separate queue that is attended by a single server. Service times are independent, exponentially distributed random variables. Server i works at rate , where μ is constant. We prove that the departure process becomes stochastically faster as the service rates become more homogeneous in the sense of stochastic majorization. Consequently, when all k servers work with equal rates the departure process is stochastically maximized.

The matrix M/M/∞ system: retrial models and Markov Modulated sources

1993 ◽  
Vol 25 (02) ◽  
pp. 453-471
Author(s):  
J. Keilson ◽  
L. D. Servi
Keyword(s):  
Numerical Evaluation ◽  
Stability Conditions ◽  
Kummer Functions ◽  
The Matrix ◽  
Service Rates ◽  
System Properties ◽  
Markov Modulated ◽  
The Stability ◽  
Analytical Expressions

The matrix-geometric work of Neuts could be viewed as a matrix variant of M/M/1. A 2 × 2 matrix counterpart of Neuts for M/M/∞ is introduced, the stability conditions are identified, and the ergodic solution is solved analytically in terms of the ten parameters that define it. For several cases of interest, system properties can be found from simple analytical expressions or after easy numerical evaluation of Kummer functions. When the matrix of service rates is singular, a qualitatively different solution is derived. Applications to telecommunications include some retrial models and an M/M/∞ queue with Markov-modulated input.

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