The interest to retrial queueing systems is due to their application to telephone systems. The paper studies multiserver retrial queueing systems with $n$ servers. Arrival process is a quite general point process. An arriving customer occupies one of free servers. If upon arrival all servers are busy, then the customer waits for his service in orbit, and after random time retries more and more to occupy a server. The orbit has one waiting space only, and arriving customer, who finds all servers busy and the waiting space occupied, losses from the system. Time intervals between possible retrials are assumed to have arbitrary distribution (the retrial scheme is exactly explained in the paper). The paper provides analysis of this system. Specifically the paper studies optimal number of servers to decrease the loss proportion to a given value. The representation obtained for loss proportion enables us to solve the problem numerically. The algorithm for numerical solution includes effective simulation, which meets the challenge of rare events problem in simulation.
The paper studies a multiserver retrial queueing system with $m$ servers. Arrival process is a point process with strictly stationary and ergodic increments. A customer arriving to the system occupies one of the free servers. If upon arrival all servers are busy, then the customer goes to the secondary queue, orbit, and after some random time retries more and more to occupy a server. A service time of each customer is exponentially distributed random variable with parameter $μ_1$. A time between retrials is exponentially distributed with parameter $μ_2$ for each customer. Using a martingale ap…
We consider a Markovian single-server retrial queueing system with a constant retrial rate. Conditions of null ergodicity and exponential ergodicity for the correspondent process, as well as bounds on the rate of convergence are obtained.
We study the stability of single server retrial queues under general distribution for retrial times and stationary ergodic service times, for three main retrial policies studied in the literature: classical linear, constant and control policies. The approach used is the renovating events approach to obtain sufficient stability conditions by strong coupling convergence of the process modeling the dynamics of the system to a unique stationary ergodic regime. We also obtain instability conditions by convergence in distribution to improper limiting sequences.
A single server retrial queueing system with two-classes of orbiting customers, and general class dependent service times is considered. If an arriving customer finds the server unavailable, it enters a virtual queue, called the orbit, according to its type. The customers from the orbits retry independently to access the server according to the constant retrial policy. We derive the generating function of the stationary distribution of the number of orbiting customers at service completion epochs in terms of the solution of a Riemann boundary value problem. For the symmetrical system we also d…
When an explicit expression for a probability distribution function $F(x)$ can not be found, asymptotic properties of the tail probability function $\bar{F}(x)=1-F(x)$ are very valuable, since they provide approximations or bounds for system performance, and approaches for computing probability distribution. In this paper, we study tail asymptotic properties for the number of the customers in the $M/G/1$ retrial queueing system. For queueing systems, studies on asymptotic tails have mainly concentrated on the first order asymptotic behavior. To best our knowledge, there is no second order tail…
Retrial phenomenon naturally arises in various systems such as call centers, cellular networks and random access protocols in local area networks. This paper gives a comprehensive survey on theory and applications of retrial queues in these systems. We investigate the state of the art of the theoretical researches including exact solutions, stability, asymptotic analyses and multidimensional models. We present an overview on retrial models arising from real world applications. Some open problems and promising research directions are also discussed.
In this paper, we analyze a retrial queueing system with Batch Markovian Arrival Processes and two types of customers. The rate of individual repeated attempts from the orbit is modulated according to a Markov Modulated Poisson Process. Using the theory of multi-dimensional asymptotically quasi-Toeplitz Markov chain, we obtain the stability condition and the algorithm for calculating the stationary state distribution of the system. Main performance measures are presented. Furthermore, we investigate some optimization problems. The algorithm for determining the optimal number of guard servers a…
We consider Markovian multiserver retrial queues where a blocked customer has two opportunities for abandonment: at the moment of blocking or at the departure epoch from the orbit. In this queueing system, the number of customers in the system (servers and buffer) and that in the orbit form a level-dependent quasi-birth-and-death (QBD) process whose stationary distribution is expressed in terms of a sequence of rate matrices. Using a simple perturbation technique and a matrix analytic method, we derive Taylor series expansion for nonzero elements of the rate matrices with respect to the number…
A classical result for the steady-state queue-length distribution of single-class queueing systems is the following: the distribution of the queue length just before an arrival epoch equals the distribution of the queue length just after a departure epoch. The constraint for this result to be valid is that arrivals, and also service completions, with probability one occur individually, i.e., not in batches. We show that it is easy to write down somewhat similar balance equations for {\em multidimensional} queue-length processes for a quite general network of multiclass multiserver queues. We f…
Stochastic networks with complex structures are key modelling tools for many important applications. In this paper, we consider a specific type of network: the retrial queueing systems with priority. This type of queueing system is important in various applications, including telecommunication and computer management networks with big data. For this type of system, we propose a detailed stochastic decomposition approach to study its asymptotic behaviour of the tail probability of the number of customers in the steady-state for retrial queues with two types (Type-1 and Type-2) of customers, in …
We consider a heterogeneous queueing system consisting of one large pool of $O(r)$ identical servers, where $r\to\infty$ is the scaling parameter. The arriving customers belong to one of several classes which determines the service times in the distributional sense. The system is heavily loaded in the Halfin-Whitt sense, namely the nominal utilization is $1-a/\sqrt{r}$ where $a>0$ is the spare capacity parameter. Our goal is to obtain bounds on the steady state performance metrics such as the number of customers waiting in the queue $Q^r(\infty)$. While there is a rich literature on deriving p…
In many-server systems it is crucial to staff the right number of servers so that targeted service levels are met. These staffing problems typically lead to constraint satisfaction problems that are hard to solve. During the last decade, a powerful many-server asymptotic theory has been developed to solve such problems and optimal staffing rules are known to obey the square-root staffing principle. This paper develops many-server asymptotics in the so-called QED regime, and presents refinements to many-server asymptotics and square-root staffing for a Markovian queueing model with admission co…
We study a vacation-type queueing model, and a single-server multi-queue polling model, with the special feature of retrials. Just before the server arrives at a station there is some deterministic glue period. Customers (both new arrivals and retrials) arriving at the station during this glue period will be served during the visit of the server. Customers arriving in any other period leave immediately and will retry after an exponentially distributed time. Our main focus is on queue length analysis, both at embedded time points (beginnings of glue periods, visit periods and switch- or vacatio…
We calculate asymptotics of the distribution of the number of customers in orbit in a two-class priority retrial $M/G/1$-type queueing model. In this model, priority customers wait in line while non-priority customers join an orbit and retry later. Although the generating function and moments of the number of customers in orbit has been analyzed before, asymptotics of the distribution have not been thoroughly investigated. We use singularity analysis of the probability generating function to do just that. Our results show that different regimes exist for these asymptotics in case of light-tail…
The presented study elaborates a multi-server priority queueing model considering the pre-emptive repeat policy and phase-type distribution (PH) for retrial process. The incoming heterogeneous calls are categorized as handoff calls and new calls. The arrival and service processes of both types of calls follow marked Markoian arrival process (MMAP) and PH distribution with distinct parameters, re-spectively. An arriving new call will be blocked when all the channels are occupied, and consequently will join the orbit (virtual space) of infinite capacity to retry following PH distribution. When a…
The presented study elaborates a multi-server catastrophic retrial queueing model considering preemptive repeat priority policy with phase-type (PH) distributed retrial times. For the sake of comprehension, the scenario of model operation prior and later to the occurrence of the disaster is referred to as the normal scenario and as the catastrophic scenario, respectively. In the normal scenario, the incoming heterogeneous calls are categorized as handoff calls and new calls. Handoff calls are provided controllable preemptive priority over new calls. In the catastrophic scenario, when a disaste…
The presented study elaborates a multi-server catastrophic retrial queueing model considering preemptive repeat priority policy with phase-type (PH) distributed retrial times. For the sake of comprehension, the scenario of model operation prior and later to the occurrence of the disaster is referred to as the normal scenario and as the catastrophic scenario, respectively. In both scenarios, the arrival and service processes of all types of calls follow marked Markovian arrival process (MMAP) and PH distribution with distinct parameters, respectively. In the normal scenario, the incoming hetero…
This work focuses over the performability analysis of a multi-server retrial queueing model with phase-type inter-retrial times in cellular networks. It is considered that the pattern of the new call arrival and handoff call arrival follows Markovian arrival process (MAP). The service times of both types of calls are phase-type (PH ) distributed with different service rates, and inter-failure & inter-repair times of channels are exponentially distributed. For the prioritization of handoff calls, G channels are kept in reserve for handoff calls. When all the available channels, say S, are busy …
In this paper one presents method for the computation of convergence bounds for four classes of multiserver queueing systems, described by inhomogeneous Markov chains. Specifically one considers inhomogeneous $M/M/S$ queueing system with possibly state-dependent arrival and service intensities and additionally possible batch arrivals and batch service. The unified approach based on logarithmic norm of linear operators for obtaining sharp upper and lower bounds on the rate of convergence and corresponding sharp perturbation bounds is described. As a side result, one shows by virtue of numerical…
