Gennadi Falin

A finite source retrial queue

This paper deals with a single-server retrial queue with a finite number of sources. Our analysis extends previous work on this topic and includes the analysis of the arriving customers distribution, the busy period and the waiting time process. This queuing system and its variants are widely used to model magnetic disk memory systems, star-like local area networks and other communication systems.

ON THE VIRTUAL WAITING TIME IN AN M/G/1 RETRIAL QUEUE

This paper deals with the stationary distribution of the virtual waiting time, i.e. the time between the arrival and the beginning of service of a customer in a single-server queue that operates as follows. If the server is busy at an arrival time, the customer is rejected. This customer attempts service again after some random delay and continues to do so until the first time at which the server is idle. At this time, the customer is served and leaves the system after service completion. Interarrival times and delays are assumed to be two independent sequences of i.i.d. exponentially distributed random variables. Service times are also i.i.d., generally distributed, and independent of the previous sequences.

The M/M/∞ queue in a random environment

We consider the M/M/∞ queueing system with arrival and service rate depending on the state of an auxiliary semi-Markov process (which can be viewed as an external environment) and find the mean number of customers in the system in steady state. In a particular case when the external environment can be only in two states we find the distribution of the number of customers in the system.

Information theoretic approximations for 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.

The M/M/1 retrial queue with retrials due to server failures

Sherman and Kharoufeh (Oper. Res. Lett. 34:697–705, [2006]) considered an M/M/1 type queueing system with unreliable server and retrials. In this model it is assumed that if the server fails during service of a customer, the customer leaves the server, joins a retrial group and in random intervals repeats attempts to get service. We suggest an alternative method for analysis of the Markov process, which describes the functioning of the system, and find the joint distribution of the server state, the number of customers in the queue and the number of customers in the retrial group in steady state.

A single-server batch arrival queue with returning customers

We consider a new class of batch arrival retrial queues. By contrast to standard batch arrival retrial queues we assume if a batch of primary customers arrives into the system and the server is free then one of the customers starts to be served and the others join the queue and then are served according to some discipline. With the help of Lyapunov functions we have obtained a necessary and sufficient condition for ergodicity of embedded Markov chain and the joint distribution of the number of customers in the queue and the number of customers in the orbit in steady state. We also have suggested an approximate method of analysis based on the corresponding model with losses.

Estimation of retrial rate in a retrial queue

We consider estimation of the rate of retrials for anM/M/1 repeated orders queueing system with the help of integral estimators. The main problem is connected with the statistical accuracy of the estimator, i.e. with its variance. We derive a simple asymptotic formula for this variance when the interval of observation is long. In connection with this problem we introduce a new Markovian description of retrial queues.

Stochastic inequalities for M/G/1 retrial queues

Consider an M/G/1 retrial queue. The performance characteristics of such a system are available in explicit form; however they are cumbersome (these formulas include integrals of Laplace transform, solutions of functional equations, etc.) In this paper we use the general theory of stochastic orderings to investigate the monotonicity properties of the system relative to the strong stochastic ordering, convex ordering and Laplace ordering. These results imply in particular simple insensitive bounds for the stationary distribution of the number of customers in the system and the mean number of customers served during a busy period.

Performance analysis of a hybrid switching system where voice messages can be queued

In the classical model of a hybrid switching system with movable boundary it is assumed that blocked voice messages are lost and do not affect the further functioning of the system. We describe a more realistic model where blocked voice messages are queued and then are served once a channel becomes free. The main mathematical difficulty in the analysis of such models lies in the fact that the underlying stochastic process has as state space the whole quadrant ℤ+2. We reduce the problem to a set of equations defined over the lattice semi-strip {1,...,N} × ℤ+. This in turn allows us to use available general mathematical theories.

Heavy traffic analysis of a random walk on a lattice semi-strip

We consider a Markov chain whose state space is a product of non-negative integers and a finite set. Transition probabilities satisfy certain conditions of a limited spacial homogeneity with respect to the first coordinate, . We investigate asymptotic behaviour of the invariant measure when the chain is “almost nonergodic”. The main result states that under this condition: (a) the scaled first component of the chain , and the second component , are asymptotically independent; (b) the scaled first component , is asymptotically exponential. The parameter of this limit distribution is given in terms of characteristics of the so-called induced chain (which describes the stochastic dynamics of and the first two moments of drift of