Poission process and related processes renewal-reward processes discrete-time Markov chains continuous-time Markov chains Markov chains and queues discrete-time Markov decision processes semi-Markov decision processes advanced renewal theory algorithms analysis of queuing models useful tools in applied probability useful probability distributions generating functions the discrete fast Fourier transform Laplace transformtheory numerical Laplace inversion the root-finding problem.

/pdf/a-first-course-in-stochastic-models-360xjbp56a.pdf

A First Course in Stochastic Models

Over recent years it has become increasingly evident that “classical” queueing theory cannot easily handle complex queueing systems and networks with many interacting elements. As a consequence, alternative ideas and tools, analogous to those applied in the field of Statistical Mechanics, have been proposed in the literature. In this context, the principles of Maximum Entropy (ME) and Minimum Relative Entropy (MRE), a generalisation, provide consistent methods of inference for characterising the form of an unknown but true probability distribution, based on information expressed in terms of known to exist true expected values or when, in addition, there exists a prior estimate of the unknown distribution. This paper traces the progress achieved so far towards the creation of ME and MRE product-form approximations and related algorithms for the performance analysis of general Queueing Network Models (QNMs) and indicates potential research extensions in the area.

Entropy maximisation and queueing network models

Retrial queueing systems are widely used in teletraffic theory and computer and communication networks. Although there has been a rapid growth in the literature on retrial queueing systems, the research on retrial queues with nonexponential retrial times is very limited. This paper is concerned with the analytical treatment of an M/G/1 retrial queue with general retrial times. Our queueing model is different from most single server retrial queueing models in several respectives. First, customers who find the server busy are queued in the orbit in accordance with an FCFS (first-come-first-served) discipline and only the customer at the head of the queue is allowed for access to the server. Besides, a retrial time begins (if applicable) only when the server completes a service rather upon a sen ice attempt failure. We carry out an extensive analysis of the queue, including a necessary and sufficient condition for the system to be stable, the steady state distribution of the server state and the orbit length, the waiting time distribution, the busy period, and other related quantities. Finally, we study the joint distribution of the server state and the orbit length in non-stationary regime.

Stochastic analysis of a single server retrial queue with general retrial times

We describe main models and results of a new branch of the queueing theory, theory of retrial queues, which is characterized by the following basic assumption: a customer who cannot get service (due to finite capacity of the system, balking, impatience, etc.) leaves the service area, but after some random delay returns to the system again. Emphasis is done on comparison with standard queues with waiting line and queues with losses. We give a survey of main results for both single server M/G/1 type and multiserver M/M/c type retrial queues and discuss similarities and differences between the retrial queues and their standard counterparts. We demonstrate that although retrial queues are closely connected with these standard queueing models they, however, possess unique distinguished features. We also mention some open problems.

/pdf/standard-and-retrial-queueing-systems-a-comparative-analysis-mdit8e9izl.pdf

Standard and retrial queueing systems: a comparative analysis

Single server retrial queues with priority calls

In teletraffic applications of retrial queues only the service zone is observable. Another part of a retrial queue, the orbit, which represents the delay before repeated attempts to get service, cannot be observed. Thus, it is very important to get general results about behavior of the orbit. We investigate two characteristics of the orbit, namely, the orbit busy period and the orbit idle period, which seem to be very useful from this point of view.

On the orbit characteristics of the M/G/1 retrial queue

In this paper we present information theoretic approximations for theM/G/1 queue with retrials. Various approximations for this model are obtained according to the available information about the service time probability density and the steady-state distribution of the system state. The results are well-suited for numerical computation.

Information theoretic approximations for the M/G /1 retrial queue

We are concerned with the M/G/1 queue with repeated attempts where a customer who finds the server busy leaves the service area and repeats his request after a random amount of time. We concentrate on the study of the waiting time process. Its analysis in terms of Laplace transforms has been discussed in the literature. However, this solution has important limitations in practice. For instance, we cannot calculate the first moments of the waiting time, W, by direct differentiation. This paper supplements the existing work and provides a direct method of computation for the second moment of W. Then the maximum entropy approach is used to estimate the true waiting time distribution.

A second order analysis of the waiting time in the M/G/1 retrial queue

The paper deals with a special bivariate Markov process which can be considered as the joint process of the channel (server) state and the number of customers in the orbit of a Markovian single server retrial system with state dependent intensities Several retrial queues, analysed in the literature, are special cases in view of the versatility
of the process Necessary and sufficient conditions for the regularity, ergodicity and recurrence of the process are given For the stationary distribution a product formula
is derived Further the busy period, the number of served customers and other related quantities are studied

On a bivariate Markov process arising in the theory of single‐server retrial queues

We consider an M/G/1 retrial queueing system with two types of calls which models a telephone switching system. In the case that arriving calls are blocked due to the channel being busy, the outgoing calls are queued in priority group whereas the incoming calls enter the retrial group in order to try service again after a random amount of time. In this paper we find the Laplace-Stieltjes transform of the distribution of the virtual waiting time for an incoming call. When the arrival rate of outgoing calls is zero, it is shown that our result is consistent with the known result for a retrial queueing system with one type of call.

https://downloads.hindawi.com/archive/1993/473409.pdf

G. Falin

Papers

On the orbit characteristics of the M/G/1 retrial queue

Information theoretic approximations for the M/G /1 retrial queue

A second order analysis of the waiting time in the M/G/1 retrial queue

On a bivariate Markov process arising in the theory of single‐server retrial queues

On the virtual waiting time for an M/G/1 retrial queue with two types of calls