Markovian Queueing Model Research Articles

Transient solution of a Markovian queuing model with heterogeneous servers and catastrophes

In this paper, we study the Markovian queuing system with heterogeneous servers and catastrophes. The customers arrive according to a Poisson process and the service times follow exponential distribution. There are a finite number of c servers with different services rates and each arriving customer requires exactly one server for its service. The queue discipline is FCFS, and the customers select the servers on fastest server first(FSF) basis. We use the generating function technique to derive the transient solution of the model in a direct way. The explicit time dependent probabilities of system size are obtained and a numerical example is presented in order to show the managerial insights of the model. Some important special cases of the model are derived and discussed. Finally, the probability that arriving customer finds system busy and average number of server busy in steady state are obtained numerically.

Transient Solution of M/M/2/N System Subjected to Catastrophe cum Restoration

In this paper, we study the distribution of the number of times that a finite capacity with equal servers Markovian queuing model catastrophic-cum-restorations reaches its capacity in time t. The occurrence of a catastrophe makes the system empty instantly but the system takes its own time to be ready to accept new customers. This time is referred to as “restoration time”. The aforesaid distribution is obtained as a marginal distribution of the joint distribution of the number of customers in the system at time t and the number of times system reaches its capacity in time t under the conditions of catastrophes and restorations.

Mathematical Modelling and Performance Analysis of Single Server Two-State Batch Arrivals and Batch Service Markovian Queue with Multiple Vacations

Here a two-state batch arrivals and batch service Markovian queue with server vacation model has been taken into consideration for its mathematical modelling and performance analysis under transient state. The present Markovian queueing model considers a single server with multiple vacations and both arrivals and services in batches of variable size. Arriving customers are processed by a single server under FIFO queue discipline and arrival process follows Poisson distribution. In addition to this, both service time and vacation time of the server follows exponential distribution. Using Laplace transforms technique, solution of governing differential-difference equations of the model has been achieved and significant performance measures of the model have been explored and some special cases have been discussed as well. Finally, some valuable conclusive observations have been found and a few suggestions for future research of the model have been focused.

A Markovian multi-server queuing model with retention of reneged customers and balking

In today’s competitive world, customer’s impatience has become a serious problem. Firms are employing a number of strategies to retain their customers. Recently, Kumar and Sharma (2013) studied a finite capacity, Markovian multi-server queuing system which deals with the retention of reneged customers. The customer’s impatience may occur in the form of balking also. In this paper, we extend the work of Kumar and Sharma (2013) by including balking to take into consideration the broader perspective of customer’s impatience. The model is solved iteratively to obtain the steady-state probabilities of system size. Some useful performance measures are derived, and the effect of the probability of retaining the reneged customers on various performance measures is studied numerically. Finally, some important queuing models are derived as special cases of this model.

A Batch Arrival Non-Markovian Queue with Three Types of Service

this paper, a batch arrival non- Markovian queueing model with three types of service is considered.The customers arrive in batches of variable size.Here a single server provides three types of service ,type 1 with probability , type 2 with probability and type 3 with probability . The service times follows general distribution . The customer may choose either type of the services.The server follows multiple vacation.Whenever the system becomes empty, the server takes a vacation and the vacation follows general distribution.On returning from vacation if the server finds no customer waiting in the system,again the server goes for vacation.The system may breakdown at random and repair time follow exponential distribution. we assume restricted admissibility of arriving batches in which not all batches are allowed to join the system at all times.The probability generating function and some probability measures of the system are also found.

Transient Solution of M[X]=G=1 With Second Optional Service, Bernoulli Schedule Server Vacation and Random Break Downs

In this model, we present a batch arrival non- Markovian queueingmodel with second optional service, subject to random break downs andBernoulli vacation. Batches arrive in Poisson stream with mean arrivalrate (> 0), such that all customers demand the rst `essential' ser-vice, wherein only some of them demand the second `optional' service.The service times of the both rst essential service and the second op-tional service are assumed to follow general (arbitrary) distribution withdistribution function B1(v) and B2(v) respectively. The server may un-dergo breakdowns which occur according to Poisson process with breakdown rate . Once the system encounter break downs it enters the re-pair process and the repair time is followed by exponential distributionwith repair rate . Also the sever may opt for a vacation accordingto Bernoulli schedule. The vacation time follows general (arbitrary)distribution with distribution function v(s). The time-dependent prob-ability generating functions have been obtained in terms of their Laplacetransforms and the corresponding steady state results have been derivedexplicitly. Also the mean queue length and the mean waiting time havebeen found explicitly.

Steady State Solution of a Catastrophic-cum-Restorative M/G/1 Queuing System Using Supplementary Variable Technique

Non- Markovian queuing models have their place in modeling the real life phenomena. In fact, their utility get enhanced when one is not able to get a particular probability distribution for either the inter-arrival times or for the service times. The service times are assumed to have general service time distribution in case of computer communication modeling. Recently, the emphasis is put on the catastrophe modeling and its applications in real situations. Keeping this in view, an M/G/1 queuing model has been developed with catastrophic and restorative effects. The steady-state solution of the model has been obtained using supplementary variable technique. Some queuing models have been obtained as particular cases of this model.

In This Issue

In This Issue

Staffing Call Centers with Impatient Customers: Refinements to Many-Server Asymptotics

In call centers it is crucial to staff the right number of agents so that the targeted service levels are met. These staffing problems typically lead to constraint satisfaction problems that are hard to solve. During the last decade, a beautiful many-server asymptotic theory has been developed to solve such problems for large call centers, and optimal staffing rules are known to obey the square-root staffing principle. This paper presents refinements to many-server asymptotics and this staffing principle for a Markovian queueing model with impatient customers.

A Survey on Dynamic Spectrum Access Techniques for Cognitive Radio

Wireless networks are characterized by fixed spectrum policy. With increasing demands for wireless communication efficiently using the spectrum resources has become an essential issue. Cognitive radio is a form of wireless communication which is used to sense the spectrum and find the free spectrum. It is used by unlicensed users without causing interference to the licensed user. Cognitive radio with the dynamic spectrum access is key technology which provides the best solution by allowing a group of Secondary users to share the radio spectrum originally allocated to the primary users. Dynamically accessing the unused spectrum is known as dynamic spectrum access (DSA) which becomes a promising approach to increase the efficiency of spectrum usage. In this paper, DSA models are discussed along with different methods such as game theory based method, a measurement-based model, network coded cognitive control channel, Markovian Queuing model, the Delay performance of threshold policies, fuzzy logic based method and spatio-temporal spectrum management model.

Queueing model of a hybrid channel with faster link subject to partial and complete failures

This paper presents a Markovian queueing model for a hybrid channel consisting of two links with different throughputs. The busy faster link is assumed to be unreliable, with possible partial and complete failures. Partial failures lead to a reduction in the service rate, while complete failure stops the service. Repairs return the faster server to a non-failed state. The problem of the optimal allocation of customers between the servers is considered. The optimality of a threshold-based policy that depends on the failure state of the faster server is proved. The dynamic behaviour of the system for the given threshold policy is described by a four-dimensional Markov process that can be treated as a QBD process with a large number of boundary states. Stationary analysis of the system is performed by means of a matrix-geometric approach, and the main performance measures are derived.

Transient Analysis of Markovian Queueing Model with Bernoulli Schedule and Multiple Working Vacations

Transient Analysis of Markovian Queueing Model with Bernoulli Schedule and Multiple Working Vacations

Computational Aspects of GPU-accelerated Sparse Matrix-Vector Multiplication for Solving Markov Models

In this article we investigate some computational aspects of GPU-accelerated matrix-vector multiplication where matrix is sparse. Particularly, we deal with sparse matrices appearing in modelling with Markovian queuing models. The model we use for research is a Markovian queuing model of a wireless device. This model describes the device’s behavior during possible channel occupation by other devices. We study the efficiency of multiplication of a sparse matrix by a dense vector with the use of an appropriate, ready-to-use GPU-accelerated mathematical library, namely CUSP. For the CUSP library we discuss data structures and their impact on the CUDA platform for the fine-grained parallel architecture of the GPU. Our aim is to find the best format for storing a sparse matrix for GPU-computation (especially one associated with the Markovian model of a wireless device). We compare the time, the performance and the speed-up for the card NVIDIA Tesla C2050 (with ECC ON). For unstructured matrices (as our Markovian matrices), we observe speed-ups (in respect to CPU-only computations) of over 8 times.

Markovian Queuing Model for Dynamic Spectrum Allocation in Centralized Architecture for Cognitive Radios

Dynamic Spectrum Allocation (DSA) is one of the latest technologies serving to cater to rising demand of spectra. Since a decade, cognitive radios (CR) are looked upon as a solution to increase spectral efficiency. However, to make CR a reality, development of air interface is a big challenge. In this paper, we have proposed a CR architecture followed by its equivalent mathematical model. We consider a centralized network where the master/controller of this ad-hoc network coordinates for spectrum allocation with the surrounding CR in the network. This CR ad-hoc network is assumed to coexist with the network of licensed users where the controller of licensed users is updated with CR coordinating engine. The mathematical model is formed of two Markovian distributed queues. One of these queues is modeled as M/M/1 and the second as M/G/S/N. Mathematical analysis of our proposal is focused to evaluate bandwidth access latency of an unlicensed user.

Transient Analysis of Two-Dimensional State Markovian Queuing Model with Multiple Working Vacations and Non-Exhaustive Service

Transient Analysis of Two-Dimensional State Markovian Queuing Model with Multiple Working Vacations and Non-Exhaustive Service

Inverse product Toeplitz preconditioners for non-Hermitian Toeplitz systems

In this paper, we first propose product Toeplitz preconditioners (in an inverse form) for non-Hermitian Toeplitz matrices generated by functions with zeros. Our inverse product-type preconditioner is of the form \(T_F T_L^{-1} T_U^{-1}\) where T F , T L , and T U are full, band lower triangular, and band upper triangular Toeplitz matrices, respectively. Our basic idea is to decompose the generating function properly such that all factors T F , T L , and T U of the preconditioner are as well-conditioned as possible. We prove that under certain conditions, the preconditioned matrix has eigenvalues and singular values clustered around 1. Then we use a similar idea to modify the preconditioner proposed in Ku and Kuo (SIAM J Sci Stat Comput 13:1470–1487, 1992) to handle the zeros in rational generating functions. Numerical results, including applications to the computation of the stationary probability distribution of Markovian queuing models with batch arrival, are given to illustrate the good performance of the proposed preconditioners.

Static and Dynamic Pricing of Excess Capacity in a Make‐to‐Order Environment

We consider a make‐to‐order manufacturer that serves two customer classes: core customers who pay a fixed negotiated price, and “fill‐in” customers who make submittal decisions based on the current price set by the firm. Using a Markovian queueing model, we determine how much the firm can gain by explicitly accounting for the status of its production facility in making pricing decisions. Specifically, we examine three pricing policies: (1) static, state‐independent pricing, (2) constant pricing up to a cutoff state, and (3) general state‐dependent pricing. We determine properties of each policy, and illustrate numerically the financial gains that the firm can achieve by following each policy as compared with simpler policies. Our main result is that constant pricing up to a cutoff state can dramatically outperform a state‐independent policy, while at the same time achieving most of the increase in revenue achievable from general state‐dependent pricing. Thus, we find that constant pricing up to a cutoff state presents an attractive tradeoff between ease of implementation and revenue gain. When the costs of policy design and implementation are taken into account, this simple heuristic may actually out‐perform general state‐dependent pricing in some settings.

Performance Analysis and Improvement Methods for Channel Resource Management Strategies of LEO–MSS With Multiparty Traffic

A novel analytical framework for the accurate and efficient evaluation of the performance of channel resource management strategies for low Earth orbit mobile satellite systems (LEO-MSSs) supporting multiparty traffic is presented. By considering a fixed channel reservation (FCR) scheme as a benchmark, an efficient and accurate analytical approach is developed for obtaining the performance of multiparty traffic under various quality-of-service (QoS) performance measure criteria. The proposed approach is based on a Markovian queuing model, and its correctness and accuracy have been verified by means of computer simulations. To improve the overall performance of LEO-MSS, two novel resource management techniques are introduced and analyzed. The first one is an efficient adaptive channel reservation (ACR) scheme, which allows priority to be given to handover requests that are generated by multiparty traffic. The second one is a new call queuing (NCQ) policy, which efficiently reduces the new call blocking probability with little impact on other system performance measures, such as call dropping probability and unsuccessful call probability. Various performance results show that when ACR is used in conjunction with NCQ, extremely low blocking and handover failure probabilities can be achieved for multiparty traffic.

Performance modeling of an Apache Web server with a dynamic pool of service processes

In the current Internet the performance of service delivery crucially depends on the proper and efficient operation of Web servers. It is determined by their software architecture and characterized by the applied processing model. Here we consider the Unix software architecture of an Apache Web server with its non-threaded multi-processing module Prefork. We propose a tractable multi-server model to approximate the performance of the load-dependent dynamic behavior of Apache's resource pool of available HTTP service processes, which has not been done before. Furthermore, we show that this Markovian queueing model can be solved by advanced matrix-geometric methods. Then the efficiently computed performance results of this analytic model are compared with measurements of a real Apache Web server. The outcome clearly indicates that our analytic model can very accurately predict the mean-value performance of Apache under the Prefork policy.

Synergistic modeling of call center operations

We synergistically apply queueing theory, integer programming, and stochastic simulation to determine an optimal staffing policy for a repair call handling center. A stationary Markovian queueing model is employed to determine minimal staffing levels for a sequence of time intervals with varying call volumes and mean handling times. These staffing requirements populate an integer program model for determining the mix of call agent shifts that will achieve service quality standards at minimum cost. Since the analytical modeling requires simplifying assumptions, expected performance of the optimal staffing policy is evaluated using stochastic simulation. Computational efficiency of the simulation is improved dramatically by employing the queueing model to generate an analytic control variate.