Performance Analysis of a Block-Structured Discrete-Time Retrial Queue with State-Dependent Arrivals

Preface to the Second Edition. Preface to the First Edition. 1. Introduction. 1.1 Motivation. 1.2 Methodological Background. 1.3 Basics of Probability and Statistics. 2. Markov Chains. 2.1 Markov Processes. 2.2 Performance Measures. 2.3 Generation Methods. 3. Steady-State Solutions of Markov Chains. 3.1 Solution for a Birth Death Process. 3.2 Matrix-Geometric Method: Quasi-Birth-Death Process. 3.3 Hessenberg Matrix: Non-Markovian Queues. 3.4 Numerical Solution: Direct Methods. 3.5 Numerical Solution: Iterative Methods. 3.6 Comparison of Numerical Solution Methods. 4. Steady-State Aggregation/Disaggregation Methods. 4.1 Courtois' Approximate Method. 4.2 Takahashi's Iterative Method. 5. Transient Solution of Markov Chains. 5.1 Transient Analysis Using Exact Methods. 5.2 Aggregation of Stiff Markov Chains. 6. Single Station Queueing Systems. 6.1 Notation. 6.2 Markovian Queues. 6.3 Non-Markovian Queues. 6.4 Priority Queues. 6.5 Asymmetric Queues. 6.6 Queues with Batch Service and Batch Arrivals. 6.7 Retrial Queues. 6.8 Special Classes of Point Arrival Processes. 7. Queueing Networks. 7.1 Definitions and Notation. 7.2 Performance Measures. 7.3 Product-Form Queueing Networks. 8. Algorithms for Product-Form Networks. 8.1 The Convolution Algorithm. 8.2 The Mean Value Analysis. 8.3 Flow Equivalent Server Method. 8.4 Summary. 9. Approximation Algorithms for Product-Form Networks. 9.1 Approximations Based on the MVA. 9.2 Summation Method. 9.3 Bottapprox Method. 9.4 Bounds Analysis. 9.5 Summary. 10. Algorithms for Non-Product-Form Networks. 10.1 Nonexponential Distributions. 10.2 Different Service Times at FCFS Nodes. 10.3 Priority Networks. 10.4 Simultaneous Resource Possession. 10.5 Prograrns with Internal Concurrency. 10.6 Parallel Processing. 10.7 Networks with Asymmetric Nodes. 10.8 Networks with Blocking. 10.9 Networks with Batch Service. 11. Discrete-Event Simulation. 11.1 Introduction to Simulation. 11.2 Simulative or Analytic Solution? 11.3 Classification of Simulation Models. 11.4 Classification of Tools in DES. 11.5 The Role of Probability and Statistics in Simulation. 11.6 Applications. 12. Performance Analysis Tools. 12.1 PEPSY. 12.2 SPNP. 12. 3 MOSEL-2. 12.4 SHARPE. 12.5 Characteristics of Some Tools. 13. Applications. 13.1 Case Studies of Queueing Networks. 13.2 Case Studies of Markov Chains. 13.3 Case Studies of Hierarchical Models. Glossary. Bibliography. Index.

A single-server retrial queue with a MAP flow, PH service times and a pool of finite capacity for accumulation of the customers and their group service is considered. Service to the next group is not provided until the number of customers in the pool will reach a certain predefined threshold value. The service time of a group depends on its size and it is less than the sum of the individual service times. The dependencies of the basic performance measures of the system on the capacity of the pool and the threshold are obtained. Numerical results are presented.

Performance analysis of statistical multiplexing of heterogeneous discrete-time Markovian arrival processes in an ATM network

The cell generation process resulting from variable bit rate (VBR) video traffic in an ATM environment is known to exhibit a statistical behavior which is not accurately captured by the standard Markovian models. In such an environment, one may identify different cycles where in each cycle, the cell rate and the cell activity process may change significantly. We introduce a new class of discrete time Markovian arrival processes referred to as cycle modulated sources (CMS) where we extend and further generalize the model for general on-off sources to allow different cycles and dependency among them. Each cycle is identified by its on and off period and its cell spacing parameter. Some results on the correlational structure of CMS is provided, where by means of examples we show that for large (but not too large) lags correlation can be easily obtained by simple CMS which, is a known property of VBR video traffic. We also examine the queue response to such sources. We find that the distribution, of cells in a queueing environment driven by such sources exhibits a very heavy tail which is not usually observed by traditional models.

Analysis of a batch-service queue with variable service capacity, correlated customer types and generally distributed class-dependent service times

Analysis of a single-server retrial queue with FCFS orbit and Bernoulli vacation

Performance analysis of the retrial queues with finite number of sources and service interruptions

In the last chapter, we used Markov process models for queueing systems with Poisson arrivals and exponential service times. To model a system as a Markov process, we should be able to give complete distribution characteristics of the process beyond time t, using what we know about the process at t and changes that may occur after t, without referring back to the events before t. When arrivals are Poisson and service times are exponential, because of the memoryless property of the exponential distribution we are able to use the Markov process as a model. If the arrival rate is λ and service rate is μ, at any time point t, time to next arrival has the exponential distribution with rate λ, and if a service is in progress, the remaining service time has the exponential distribution with rate μ. If one or both of the arrival and service distributions are non-exponential, the memoryless property does not hold and a Markov model of the type discussed in the last chapter does not work. In this chapter, we discuss a method by which a Markov model can be constructed, not for all t, but for specific time points on the time axis.KeywordsMarkov ChainService TimeTransition Probability MatrixProbability Generate FunctionService Time DistributionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the present paper, a single server retrial queue with impatient customers is studied. The primary arrivals and repeating calls follow the Poisson distribution. The service time is exponentially distributed. Explicit time-dependent probabilities of an exact number of arrivals and departures from the orbit are obtained by solving the differential-difference equations recursively. Steady state solution of the number of busy servers is obtained. The numerical results are graphically displayed to illustrate the effect of arrival rate, retrial rate and service rate on different probabilities against time. Some special cases of interest are also deduced.

Customer retrials commonly occur in many service systems, such as healthcare, call centers, mobile networks, computer systems, and inventory systems. However, because of their complex nature, retrial queues are often more difficult to analyze than queues without retrials. In “Reducing Delay in Retrial Queues by Simultaneously Differentiating Service and Retrial Rates”, J. Wang, Z. Wang, and Y. Liu develop a service grade differentiation policy for queueing models with customer retrials. They show that the average waiting time can be reduced through strategically allocating the rates of service and retrial times without needing additional service capacity. Counter to the intuition that higher service variability usually yields a larger delay, the authors show that the benefits of this simultaneous service-and-retrial differentiation (SSRD) policy outweigh the impact of the increased service variability. To validate the effectiveness of the new SSRD policy, the authors provide (i) conditions under which SSRD is more beneficial, (ii) closed-form expressions of the optimal policy, (iii) asymptotic reduction of customer delays when the system is in heavy traffic, and (iv) insightful observations/discussions and numerical results.

In this paper we study a broad class of semi-Markovian queues introduced by Sengupta. This class contains many classical queues such as the GI/M/1 queue, SM/MAP/1 queue and others, as well as queues with correlated inter-arrival and service times. Queues belonging to this class are characterized by a set of matrices of size m and Sengupta showed that its waiting time distribution can be represented as a phase-type distribution of order m. For the special case of the SM/MAP/1 queue without correlated service and inter-arrival times the queue length distribution was also shown to be phase-type of order m, but no derivation for the queue length was provided in the general case. This paper introduces an order m^2 phase-type representation (kappa, K) for the queue length distribution in the general case. Moreover, we prove that the order m^2 of the distribution cannot be further reduced in general. Examples for which the order is between m and m^2 are also identified. We derive these results in both discrete and continuous time and also discuss the numerical procedure to compute (kappa, K). Moreover, by combining a result of Sengupta and Ozawa, we provide a simple formula to compute the order m phase-type representation of the waiting time in a MAP/MAP/1 queue without correlated service and inter-arrival times, using the R matrix of a Quasi-Birth-Death Markov chain.

In this paper, we study a multi-server queueing system with retrials and an infinite orbit. The arrival of primary customers is described by a batch Markovian arrival process (BMAP), and the service times have a phase-type (PH) distribution. Previously, in the literature, such a system was mainly considered under the strict assumption that the intervals between the repeated attempts from the orbit have an exponential distribution. Only a few publications dealt with retrial queueing systems with non-exponential inter-retrial times. These publications assumed either the rate of retrials is constant regardless of the number of customers in the orbit or this rate is constant when the number of orbital customers exceeds a certain threshold. Such assumptions essentially simplify the mathematical analysis of the system, but do not reflect the nature of the majority of real-life retrial processes. The main feature of the model under study is that we considered the classical retrial strategy under which the retrial rate is proportional to the number of orbital customers. However, in this case, the assumption of the non-exponential distribution of inter-retrial times leads to insurmountable computational difficulties. To overcome these difficulties, we supposed that inter-retrial times have a phase-type distribution if the number of customers in the orbit is less than or equal to some non-negative integer (threshold) and have an exponential distribution in the contrary case. By appropriately choosing the threshold, one can obtain a sufficiently accurate approximation of the system with a PH distribution of the inter-retrial times. Thus, the model under study takes into account the realistic nature of the retrial process and, at the same time, does not resort to restrictions such as a constant retrial rate or to rough truncation methods often applied to the analysis of retrial queueing systems with an infinite orbit. We describe the behavior of the system by a multi-dimensional Markov chain, derive the stability condition, and calculate the steady-state distribution and the main performance indicators of the system. We made sure numerically that there was a reasonable value of the threshold under which our model can be served as a good approximation of the BMAP/PH/N queueing system with the PH distribution of inter-retrial times. We also numerically compared the system under consideration with the corresponding queueing system having exponentially distributed inter-retrial times and saw that the latter is a poor approximation of the system with the PH distribution of inter-retrial times. We present a number of illustrative numerical examples to analyze the behavior of the system performance indicators depending on the system parameters, the variance of inter-retrial times, and the correlation in the input flow.

This chapter is concerned with the analysis of a single server retrial queue with two types of service, Bernoulli vacation and feedback. The server provides two types of service i.e., type 1 service with probability??1 and type 2 service with probability ??2. We assume that the arriving customer who finds the server busy upon arrival leaves the service area and are queued in the orbit in accordance with an FCFS discipline and repeats its request for service after some random time. After completion of type 1 or type 2 service the unsatisfied customer can feedback and joins the tail of the retrial queue with probability f or else may depart from the system with probability 1–f. Further the server takes vacation under Bernoulli schedule mechanism, i.e., after each service completion the server takes a vacation with probability q or with probability p waits to serve the next customer. For this queueing model, the steady state distributions of the server state and the number of customers in the orbit are obtained using supplementary variable technique. Finally the average number of customers in the system and average number of customers in the orbit are also obtained.

A novel retrial multi-server queueing model with batch arrival of customers is considered. In contrast to the standard batch arrival when a whole batch arrives into the system at one epoch, we assume that the customers of a batch arrive one-by one in exponentially distributed times. Service time is exponentially distributed. Flow of batches is the stationary Poisson arrival process. Batch size distribution is geometric. The number of batches, which can be admitted into the system simultaneously, is subject of control. Analysis of the joint distribution of the number batches and customers in the system is implemented using the tool of the multi-dimensional asymptotically quasi-Toeplitz Markov chains.

We consider the susceptible-infective (SI) epidemiological model, a variant of the Kermack-McKendrick models, and let the contact rate be a function of the number of infectives, an indicator of disease spread during the course of the epidemic. We represent the resultant model as a continuous-time Markov chain. The result is a pure death (or birth) process with state-dependent rates, for which we find the probability distribution of the associated Markov chain by solving the Kolmogorov forward equations. This model is used to find the analytic solution of the SI model as well as the distribution of the epidemic duration. We use the maximum likelihood method to estimate contact rates based on observations of inter-infection time intervals. We compare the stochastic model to the corresponding deterministic models through a numerical experiment within a typical household. We also incorporate different intervention policies for vaccination, antiviral prophylaxis, isolation, and treatment considering both full and partial adherence to interventions among individuals.