Multichannel queueing systems with infinite waiting room and stochastic control

1989 ◽  
Vol 26 (2) ◽  
pp. 345-362 ◽  
Author(s):  
Jewgeni Dshalalow
Keyword(s):  
Queueing Systems ◽  
Invariant Probability Measure ◽  
Transportation Systems ◽  
Regenerative Processes ◽  
Input Stream ◽  
Waiting Room ◽  
Queueing Process ◽  
Servicing Process ◽  
Virtual Waiting Time ◽  
Necessary And Sufficient

A wide class of multichannel queueing models appears to be useful in practice where the input stream of customers can be controlled at the moments preceding the customers' departures from the source (e.g. airports, transportation systems, inventories, tandem queues). In addition, the servicing facility can govern the intensity of the servicing process that further improves flexibility of the system. In such a multichannel queue with infinite waiting room the queueing process {Zt; t ≧ 0} is under investigation. The author obtains explicit formulas for the limiting distribution of (Zt) partly using an approach developed in previous work and based on the theory of semi-regenerative processes. Among other results the limiting distributions of the actual and virtual waiting time are derived. The input stream (which is not recurrent) is investigated, and distribution of the residual time from t to the next arrival is obtained. The author also treats a Markov chain embedded in (Zt) and gives a necessary and sufficient condition for its existence. Under this condition the invariant probability measure is derived.

Multichannel queueing systems with infinite waiting room and stochastic control

1989 ◽  
Vol 26 (02) ◽  
pp. 345-362
Author(s):  
Jewgeni Dshalalow
Keyword(s):  
Queueing Systems ◽  
Invariant Probability Measure ◽  
Transportation Systems ◽  
Regenerative Processes ◽  
Input Stream ◽  
Waiting Room ◽  
Queueing Process ◽  
Servicing Process ◽  
Virtual Waiting Time ◽  
Necessary And Sufficient

A wide class of multichannel queueing models appears to be useful in practice where the input stream of customers can be controlled at the moments preceding the customers' departures from the source (e.g. airports, transportation systems, inventories, tandem queues). In addition, the servicing facility can govern the intensity of the servicing process that further improves flexibility of the system. In such a multichannel queue with infinite waiting room the queueing process {Zt ; t ≧ 0} is under investigation. The author obtains explicit formulas for the limiting distribution of (Zt ) partly using an approach developed in previous work and based on the theory of semi-regenerative processes. Among other results the limiting distributions of the actual and virtual waiting time are derived. The input stream (which is not recurrent) is investigated, and distribution of the residual time from t to the next arrival is obtained. The author also treats a Markov chain embedded in (Zt ) and gives a necessary and sufficient condition for its existence. Under this condition the invariant probability measure is derived.

Comparing multi-server queues with finite waiting rooms, II: Different numbers of servers

1979 ◽  
Vol 11 (02) ◽  
pp. 448-455 ◽  
Author(s):  
David Sonderman
Keyword(s):  
Probability Space ◽  
Queueing Systems ◽  
Stochastic Order ◽  
Sample Path ◽  
System Size ◽  
Waiting Room ◽  
Virtual Waiting Time ◽  
Number Of Customers ◽  
Multi Server ◽  
Quantities Of Interest

We compare two queueing systems with identical general arrival streams, but different numbers of servers, different waiting room capacities, and stochastically ordered service time distributions. Under appropriate conditions, it is possible to construct two new systems on the same probability space so that the new systems are probabilistically equivalent to the original systems and each sample path of the stochastic process representing system size in one system lies entirely below the corresponding sample path in the other system. This construction implies stochastic order for these processes and many associated quantities of interest, such as a busy period, the number of customers lost in any interval, and the virtual waiting time.

Comparing multi-server queues with finite waiting rooms, II: Different numbers of servers

1979 ◽  
Vol 11 (2) ◽  
pp. 448-455 ◽  
Author(s):  
David Sonderman
Keyword(s):  
Probability Space ◽  
Queueing Systems ◽  
Stochastic Order ◽  
Sample Path ◽  
System Size ◽  
Waiting Room ◽  
Virtual Waiting Time ◽  
Number Of Customers ◽  
Multi Server ◽  
Quantities Of Interest

We compare two queueing systems with identical general arrival streams, but different numbers of servers, different waiting room capacities, and stochastically ordered service time distributions. Under appropriate conditions, it is possible to construct two new systems on the same probability space so that the new systems are probabilistically equivalent to the original systems and each sample path of the stochastic process representing system size in one system lies entirely below the corresponding sample path in the other system. This construction implies stochastic order for these processes and many associated quantities of interest, such as a busy period, the number of customers lost in any interval, and the virtual waiting time.

scholarly journals On a multi-channel queue with state dependent input flow and interruptions

1989 ◽  
Vol 2 (3) ◽  
pp. 199-204
Author(s):  
Jewgeni Dshalalow
Keyword(s):  
Queueing System ◽  
Input Flow ◽  
Regenerative Processes ◽  
Waiting Room ◽  
New Techniques ◽  
Queueing Process ◽  
State Dependent ◽  
Flow Activity ◽  
The Given ◽  
Explicit Relation

This paper deals with a multi-channel queueing system with a finite waiting room but without losses. The latter is achieved by a temporary interruption of the input flow activity until the waiting room is ready to place a new customer. In addition, the input flow on its “busy period” is non-recurrent: It is state dependent and may be controlled over relevant times of decision making. A similar model without interruptions (i.e. with losses) was earlier studied by the author, where in particular, major probability characteristics of the queueing process in equilibrium were obtained. Now the author derives a simple explicit relation between the two models allowing the given queue to inherit the results previously obtained. New techniques for semi-regenerative processes are used.

OPTIMIZATION OF MARKOV SYSTEMS OF QUEUEING WITH WAITING TIME IN THE MATLAB SYSTEM

2017 ◽  
pp. 39-47
Author(s):  
Viktor Afonin ◽  
Vladimir Valer'evich Nikulin
Keyword(s):  
Queueing System ◽  
Queueing Systems ◽  
Random Variable ◽  
Model Systems ◽  
Queuing Systems ◽  
Input Flow ◽  
Continuous Random Variable ◽  
Input Stream ◽  
Time Intervals ◽  
Markov Systems

The article focuses on attempt to optimize two well-known Markov systems of queueing: a multichannel queueing system with finite storage, and a multichannel queueing system with limited queue time. In the Markov queuing systems, the intensity of the input stream of requests (requirements, calls, customers, demands) is subject to the Poisson law of the probability distribution of the number of applications in the stream; the intensity of service, as well as the intensity of leaving the application queue is subject to exponential distribution. In a Poisson flow, the time intervals between requirements are subject to the exponential law of a continuous random variable. In the context of Markov queueing systems, there have been obtained significant results, which are expressed in the form of analytical dependencies. These dependencies are used for setting up and numerical solution of the problem stated. The probability of failure in service is taken as a task function; it should be minimized and depends on the intensity of input flow of requests, on the intensity of service, and on the intensity of requests leaving the queue. This, in turn, allows to calculate the maximum relative throughput of a given queuing system. The mentioned algorithm was realized in MATLAB system. The results obtained in the form of descriptive algorithms can be used for testing queueing model systems during peak (unchanged) loads.

scholarly journals Complex-analytic and matrix-analytic solutions for a queueing system with group service controlled by arrivals

2000 ◽  
Vol 13 (4) ◽  
pp. 415-427
Author(s):  
Lev Abolnikov ◽  
Alexander Dukhovny
Keyword(s):  
Block Size ◽  
Iteration Process ◽  
Analytic Solutions ◽  
Analytic Method ◽  
Service Time Distribution ◽  
Matrix Analytic Method ◽  
Queueing Process ◽  
The Matrix ◽  
Matrix Iteration ◽  
Necessary And Sufficient

A bulk M/G/1 system is considered that responds to large increases (decreases) of the queue during the service act by alternating between two service modes. The switching rule is based on two “up” and “down” thresholds for total arrivals over the service act. A necessary and sufficient condition for the ergodicity of a Markov chain embedded into the main queueing process is found. Both complex-analytic and matrix-analytic solutions are obtained for the steady-state distribution. Under the assumption of the same service time distribution in both modes, a combined complex-matrix-analytic method is introduced. The technique of “matrix unfolding” is used, which reduces the problem to a matrix iteration process with the block size much smaller than in the direct application of the matrix-analytic method.

Some analyses on the control of queues using level crossings of regenerative processes

1980 ◽  
Vol 17 (3) ◽  
pp. 814-821 ◽  
Author(s):  
J. G. Shanthikumar
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Queueing Systems ◽  
Density Functions ◽  
Waiting Time Distribution ◽  
Regenerative Processes ◽  
Stieltjes Transform ◽  
Expected Number ◽  
Control Of Queues ◽  
Special Case

Some properties of the number of up- and downcrossings over level u, in a special case of regenerative processes are discussed. Two basic relations between the density functions and the expected number of upcrossings of this process are derived. Using these reults, two examples of controlled M/G/1 queueing systems are solved. Simple relations are derived for the waiting time distribution conditioned on the phase of control encountered by an arriving customer. The Laplace-Stieltjes transform of the distribution function of the waiting time of an arbitrary customer is also derived for each of these two examples.

On queueing systems by retrials

1983 ◽  
Vol 20 (02) ◽  
pp. 380-389 ◽  
Author(s):  
Vidyadhar G. Kulkarni
Keyword(s):  
General Result ◽  
Service Time ◽  
Queueing Systems ◽  
Single Server ◽  
Expected Number ◽  
Waiting Room ◽  
Second Moments ◽  
Different Types ◽  
General Distributions ◽  
Arbitrary Service Time

A general result for queueing systems with retrials is presented. This result relates the expected total number of retrials conducted by an arbitrary customer to the expected total number of retrials that take place during an arbitrary service time. This result is used in the analysis of a special system where two types of customer arrive in an independent Poisson fashion at a single-server service station with no waiting room. The service times of the two types of customer have independent general distributions with finite second moments. When the incoming customer finds the server busy he immediately leaves and tries his luck again after an exponential amount of time. The retrial rates are different for different types of customers. Expressions are derived for the expected number of retrial customers of each type.

The busy period of the E k/G/1 queue by finite waiting room

1975 ◽  
Vol 7 (02) ◽  
pp. 416-430
Author(s):  
A. L. Truslove
Keyword(s):  
Markov Chain ◽  
Busy Period ◽  
Waiting Room ◽  
Queueing Process ◽  
Number Of Customers

For the E k /G/1 queue with finite waiting room the phase technique is used to analyse the Markov chain imbedded in the queueing process at successive instants at which customers complete service, and the distribution of the busy period, together with the number of customers who arrive, and the number of customers served, during that period, is obtained. The limit as the size of the waiting room becomes infinite is found.

PRODUCT-FORM MARKOVIAN QUEUEING SYSTEMS WITH MULTIPLE RESOURCES

2019 ◽  
pp. 1-9 ◽  
Author(s):  
Valeriy Naumov ◽  
Konstantin Samouylov
Keyword(s):  
Queueing Systems ◽  
Sufficient Conditions ◽  
Jump Process ◽  
Product Form ◽  
Temporary Increase ◽  
Stationary Probability Distribution ◽  
Markov Jump ◽  
Multiple Resources ◽  
Resource Requirements ◽  
Necessary And Sufficient

In the paper, we study general Markovian models of loss systems with random resource requirements, in which customers at arrival occupy random quantities of various resources and release them at departure. Customers may request negative quantities of resources, but total amount of resources allocated to customers should be nonnegative and cannot exceed predefined maximum levels. Allocating a negative volume of a resource to a customer leads to a temporary increase in its volume in the system. We derive necessary and sufficient conditions for the product-form of the stationary probability distribution of the Markov jump process describing the system.

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