scholarly journals On some queue length controlled stochastic processes

1990 ◽  
Vol 3 (4) ◽  
pp. 227-244 ◽  
Author(s):  
Lev Abolnikov ◽  
Jewgeni E. Dshalalow ◽  
Alexander M. Dukhovny
Keyword(s):  
Continuous Time ◽  
Queue Length ◽  
Invariant Probability Measure ◽  
Time Parameter ◽  
Batch Size ◽  
Single Server ◽  
Input Output ◽  
Time Service ◽  
Queueing Processes ◽  
Embedded Process

The authors study the input, output and queueing processes in a general controlled single-server bulk queueing system. It is supposed that inter-arrival time, service time, batch size of arriving units and the capacity of the server depend on the queue length.The authors establish an ergodicity criterion for both the queueing process with continuous time parameter and the embedded process, study their transient and steady state behavior and prove ergodic theorems for some functionals of the input, output and queueing processes. The following results are obtained: Invariant probability measure of the embedded process, stationary distribution of the process with continuous time parameter, expected value of a busy period, rates of input and output processes and the relative speed of convergence of the expected queue length. Various examples (including an optimization problem) illustrate methods developed in the paper.

scholarly journals On applications of excess level processes to (N,D)-policy bulk queueing systems

1996 ◽  
Vol 9 (4) ◽  
pp. 551-562 ◽  
Author(s):  
Jewgeni H. Dshalalow
Keyword(s):  
Continuous Time ◽  
Queueing Systems ◽  
Time Parameter ◽  
Counting Processes ◽  
Two Dimensional ◽  
Stationary Distributions ◽  
Compound Poisson ◽  
Server Vacations ◽  
Poisson Input ◽  
Queueing Processes

The paper deals with queueing systems in which N- and D-policies are combined into one. This means that an idle or vacationing server will resume his service if the queueing or workload process crosses some specified fixed level N or D, respectively. For the proposed (N,D)-policy we study the queueing processes in models with and without server vacations, with compound Poisson input, and with generally distributed service and vacation periods. The analysis of the models is essentially based on fluctuation techniques for two-dimensional marked counting processes newly developed by the author. The results enable us to arrive at stationary distributions for the embedded and continuous time parameter queueing processes in closed analytic forms, enhancing the well-known Kendall formulas and their modifications.This article is dedicated to the memory of Roland L. Dobrushin.

Stationary queue-length and waiting-time distributions in single-server feedback queues

1984 ◽  
Vol 16 (2) ◽  
pp. 437-446 ◽  
Author(s):  
Ralph L. Disney ◽  
Dieter König ◽  
Volker schmidt
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Arbitrary Point ◽  
Single Server ◽  
Waiting Room ◽  
Bernoulli Feedback ◽  
Stationary Queue Length ◽  
Queueing Processes ◽  
Number Of Customers ◽  
Feedback Time

For M/GI/1/∞ queues with instantaneous Bernoulli feedback time- and customer-stationary characteristics of the number of customers in the system and of the waiting time are investigated. Customer-stationary characteristics are thereby obtained describing the behaviour of the queueing processes, for example, at arrival epochs, at feedback epochs, and at times at which an arbitrary (arriving or fed-back) customer enters the waiting room. The method used to obtain these characteristics consists of simple relationships between them and the time-stationary distribution of the number of customers in the system at an arbitrary point in time. The latter is obtained from the wellknown Pollaczek–Khinchine formula for M/GI/1/∞ queues without feedback.

Queueing processes in bulk systems under the D-policy

1998 ◽  
Vol 35 (04) ◽  
pp. 976-989
Author(s):  
Jewgeni H. Dshalalow
Keyword(s):  
Continuous Time ◽  
Three Dimensional ◽  
Time Parameter ◽  
Counting Processes ◽  
Queueing Process ◽  
Single Vacation ◽  
Multiple Vacation ◽  
Policy Models ◽  
Queueing Processes ◽  
General Service

This paper studies the queueing process in a class of D-policy models with Poisson bulk input, general service time, and four different vacation scenarios, among them a multiple vacation, single vacation and idle server. The D-policy specifies a busy period discipline, which requires an idle or vacationing server to resume his service when the workload process crosses some fixed level D. The analysis of the queueing process is based on the theory of fluctuations for three-dimensional marked counting processes presented in the paper. For all models, we derive the stationary distributions for the embedded and continuous time parameter queueing processes in closed analytic forms and illustrate the results by a number of examples and applications.

scholarly journals The Correlation Coefficients of the Queue Lengths of Some Stationary Single Server Queues

1971 ◽  
Vol 12 (1) ◽  
pp. 35-46 ◽  
Author(s):  
A. G. Pakes
Keyword(s):  
Queue Length ◽  
Final Section ◽  
Waiting Times ◽  
Queueing Systems ◽  
Correlation Coefficients ◽  
Second Order ◽  
Single Server ◽  
Bulk Service ◽  
Queue Lengths ◽  
Queueing Processes

Until recently there has been little systematic work on the second-order properties of queueing processes. The aim of this paper is to study systematically the second-order properties of the queue length processes embedded at departure epochs in the M/G/1 and bulk service M/G/1 queues, and at arrival epochs in the GI/M/1 queue. In the latter case our results extend those of Daley [7], while in the ordinary M/G/1 queue our work parallels Daley's [6] discussion of waiting times in the same system. In the final section we briefly discuss two discrete time queueing systems.

Queueing processes in bulk systems under the D-policy

1998 ◽  
Vol 35 (4) ◽  
pp. 976-989 ◽  
Author(s):  
Jewgeni H. Dshalalow
Keyword(s):  
Continuous Time ◽  
Three Dimensional ◽  
Time Parameter ◽  
Counting Processes ◽  
Queueing Process ◽  
Single Vacation ◽  
Multiple Vacation ◽  
Policy Models ◽  
Queueing Processes ◽  
General Service

This paper studies the queueing process in a class of D-policy models with Poisson bulk input, general service time, and four different vacation scenarios, among them a multiple vacation, single vacation and idle server. The D-policy specifies a busy period discipline, which requires an idle or vacationing server to resume his service when the workload process crosses some fixed level D. The analysis of the queueing process is based on the theory of fluctuations for three-dimensional marked counting processes presented in the paper. For all models, we derive the stationary distributions for the embedded and continuous time parameter queueing processes in closed analytic forms and illustrate the results by a number of examples and applications.

Stationary queue-length and waiting-time distributions in single-server feedback queues

1984 ◽  
Vol 16 (02) ◽  
pp. 437-446 ◽  
Author(s):  
Ralph L. Disney ◽  
Dieter König ◽  
Volker schmidt
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Arbitrary Point ◽  
Single Server ◽  
Waiting Room ◽  
Bernoulli Feedback ◽  
Stationary Queue Length ◽  
Queueing Processes ◽  
Number Of Customers ◽  
Feedback Time

For M/GI/1/∞ queues with instantaneous Bernoulli feedback time- and customer-stationary characteristics of the number of customers in the system and of the waiting time are investigated. Customer-stationary characteristics are thereby obtained describing the behaviour of the queueing processes, for example, at arrival epochs, at feedback epochs, and at times at which an arbitrary (arriving or fed-back) customer enters the waiting room. The method used to obtain these characteristics consists of simple relationships between them and the time-stationary distribution of the number of customers in the system at an arbitrary point in time. The latter is obtained from the wellknown Pollaczek–Khinchine formula for M/GI/1/∞ queues without feedback.

scholarly journals A single-server queue with random accumulation level

1991 ◽  
Vol 4 (3) ◽  
pp. 203-210 ◽  
Author(s):  
Jewgeni H. Dshalalow
Keyword(s):  
Continuous Time ◽  
Queueing System ◽  
Time Parameter ◽  
Single Server ◽  
Single Server Queue ◽  
Explicit Formulas ◽  
Stationary Distributions ◽  
Random Size ◽  
Queueing Process ◽  
Server Queue

The author studies the queueing process in a single-server bulk queueing system. Upon completion of a previous service, the server can take a group of random size from customers that are available. Or, the server can wait until the queue attains a desired level. The author establishes an ergodicity criterion for both the queueing process with continuous time parameter and the imbedded process. Under this criterion, the author obtains explicit formulas for the stationary distributions of both processes by using semi-regenerative techniques.

State-dependent signalling in queueing networks

1994 ◽  
Vol 26 (02) ◽  
pp. 436-455 ◽  
Author(s):  
W. Henderson ◽  
B. S. Northcote ◽  
P. G. Taylor
Keyword(s):  
Queueing Networks ◽  
State Dependence ◽  
Product Form ◽  
Batch Size ◽  
Single Server ◽  
Negative Customers ◽  
Affect Control ◽  
State Dependent ◽  
Special Cases ◽  
Equilibrium Distributions

It has recently been shown that networks of queues with state-dependent movement of negative customers, and with state-independent triggering of customer movement have product-form equilibrium distributions. Triggers and negative customers are entities which, when arriving to a queue, force a single customer to be routed through the network or leave the network respectively. They are ‘signals' which affect/control network behaviour. The provision of state-dependent intensities introduces queues other than single-server queues into the network. This paper considers networks with state-dependent intensities in which signals can be either a trigger or a batch of negative customers (the batch size being determined by an arbitrary probability distribution). It is shown that such networks still have a product-form equilibrium distribution. Natural methods for state space truncation and for the inclusion of multiple customer types in the network can be viewed as special cases of this state dependence. A further generalisation allows for the possibility of signals building up at nodes.

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