Matrix-geometric Method Research Articles

Performance analysis of distributed video-on-demand (VoD) systems

In this paper, the performance of a distributed video-on-demand (VoD) system with multiple processing nodes is considered. Threshold-based multi-server queuing model with hysteresis is developed for studying the behaviour of the VoD system, wherein the video files are treated as the servers. The copies of video files (replicas) are assumed to be created and removed dynamically from the nodes according to certain thresholds. We provide an analytical solution for the queue size distribution by using the matrix geometric method. Analytical expressions for various performance measures, viz. average number of user-requests in the system, expected number of video-replicas in use, throughput of the VoD system and average delay in the transmission of the file, etc. are obtained. Various performance measures are also evaluated by using adaptive neuro-fuzzy inference systems (ANFIS). Numerical illustrations are provided to validate the analytical results and to compare the analytical and ANFIS results.

A heterogeneous two-server network system with balking and a Bernoulli vacation schedule

In this paper, we study a two-server Markovian network system with balking and a Bernoulli schedule under a single vacation policy, where servers have different service rates. After every service, only one server may take a vacation or continue to stay in the system. The vacation time follows an exponential distribution. An arriving customer finding both servers free will choose the faster server. If the customer finds only one server is free, this customer chooses this free server. If the customer finds both servers are not free, then this customer may join the system or balk. For this system, we obtain the steady state condition, the stationary distribution of the number of customers in the system, and the mean system size by using a matrix-geometric method. Some special cases are deduced, which match with earlier exiting results. Extensive numerical illustrations are provided. Motivation for this system model also comes from some computer communication networks with different types of traffic such as real-time traffic and non-real-time traffic, where messages can be processed by two channels (servers) with different transmission rates. The behavior of abandoning messages can be equated with the balking of customers in this system model.

A queueing model of delayed product differentiation

We consider a production system in which a supplier produces semi-finished items on a make-to-stock basis for a manufacturer that will customize the items on a make-to-order basis. The proportion of total processing time undertaken by the supplier determines how suitable the semi-finished items will be to meet customer demand. The manufacturer wishes to determine the optimal point of differentiation (the proportion of processing completed by the supplier) and its optimal semi-finished goods buffer size. We use matrix geometric methods to evaluate various performance measures for this system, and then, with enumeration techniques, obtain optimal solutions. We find that delayed product differentiation is attractive when the manufacturer can balance the costs of customer order fulfillment delay with the costs associated with unsuitable items.

An efficient computation algorithm for a multiserver feedback retrial queue with a large queueing capacity

Kumar et al. consider the M/M/c/N+c feedback queue with constant retrial rate [1]. They provide a solution for the steady state probabilities based on the matrix-geometric method. We show that there exists a more efficient computation method to calculate the steady state probabilities when N + c is large. We prove that the number of zero-eigenvalues of the characteristic matrix polynomial associated with the balance equation is ⌊ ( N + c + 2 ) / 2 ⌋ . As consequence, the remaining eigenvalues inside the unit circle can be computed in a quick manner based on the Sturm sequences. Therefore, the steady state probabilities can be determined in an efficient way.

Queuing Analysis of HCCA for Multi-Rate Wireless LANs with Truncated ARQ Protocol

In this paper, a novel queuing analytical framework is introduced to analyze the performance of the HCF controlled channel access designed as a medium access mechanism in wireless local area networks. The queuing model defines a multi-rate medium using adaptive modulation and coding which is a key point in wireless networks to increase effective transmission rate. Based on a MAP (Markovian arrival process)/PH (phase type)/1 queue with vacation and time-limited service, the model considers a limited node buffer and non-ideal channel with the automatic repeat request protocol. Using a MAP for traffic arrival process and PH distribution for packet service process, the inclusion of the non-transmission and vacation processes makes our analysis very general and comprehensive to support various types of practical traffic streams. To formulate the model, the powerful matrix-geometric method is employed, and the mutual impacts of some parameters in the MAC and the PHY layers are then evaluated to improve system performance. Finally, optimization of parameter settings to maximize throughput is discussed.

Solving Semi-Open Queuing Networks

A semi-open queuing network (SOQN) is a special type of a queuing network consisting of two parts: an inner network with a population constraint and an external queue to accommodate jobs whose entrance is delayed. We first study an SOQN with a single class of jobs in tandem configuration and then extend our study to multiclass configurations. Multiclass SOQNs fall into two categories: general pallet and dedicated pallet SOQNs. For the general pallet case, we aggregate all classes and solve the resulting single-class SOQN. For the dedicated pallet case, we construct a method based on an existing product-form approximation method for general, multiclass closed networks. Our approximation method combines the matrix-geometric method with the decomposition-aggregation approach. Numerical results show that our approximations have desirable accuracy and efficiency.

An explicit analysis of the lead time syndrome: stability condition and performance evaluation

Updating planned lead times in response to changing workload levels leads to erratic ordering behaviour, resulting in even larger variability in the workload levels and flow times. This phenomenon is called lead time syndrome, and describes the cyclic interaction between adaptive planned lead times and order sizes. Although it has been conceptually defined and intuitively accepted, formal analysis with analytical evaluation of the phenomenon has not been conducted. The objective of this paper is to provide a stronger understanding of the lead time syndrome, and to give new insights into the effects of frequently updating planned lead times. We develop a two-dimensional Markov process to model a single-item production process with orders released sensitive to the planned lead time. Using matrix-geometric methods, analytical results on the utilization level and the variability in the system are presented in relation to the frequency of updating the planned lead time. Although the average utilization level is always retained, the lead time syndrome causes an increase in the average workload level and the actual flow times of the completed orders. The variability of the planned lead time increases with the update frequency except at the utilization boundaries, where the relative effect of the update frequency diminishes.

The M/M/1 queue with single working vacation and set-up times

Using quasi birth and death process and matrix-geometric solution method, an M/M/1 queue with single working vacation and set-up times is discussed in this paper. The distributions for the stationary queue length and waiting time of a customer in the system are derived. Furthermore, stochastic decomposition structures of stationary indices are obtained, and the distributions of the additional queue length and additional delay are gained. Finally, several special cases and numerical examples demonstrate the results.

Optimal policy for bulk queue with multiple types of server breakdown

In this paper, an unreliable server MX/M/1 queueing system with multiple types of server breakdowns under N-policy is considered. Breakdowns occur only when the server is in busy state and each type of breakdown requires a random number of finite stages of repair. It is assumed that the lifetimes and phase repair time of the server for the individual stage are exponentially distributed. Matrix-geometric method is employed to obtain the mean queue length and other system characteristics. We perform extensive numerical experiment to demonstrate the effect of various parameters on the system performance characteristics.

Analysis of stationary discrete-time GI/D-MSP/1 queue with finite and infinite buffers

This paper considers a single-server queueing model with finite and infinite buffers in which customers arrive according to a discrete-time renewal process. The customers are served one at a time under discrete-time Markovian service process (D-MSP). This service process is similar to the discrete-time Markovian arrival process (D-MAP), where arrivals are replaced with service completions. Using the imbedded Markov chain technique and the matrix-geometric method, we obtain the system-length distribution at a prearrival epoch. We also provide the steady-state system-length distribution at an arbitrary epoch by using the supplementary variable technique and the classical argument based on renewal-theory. The analysis of actual-waiting-time (in the queue) distribution (measured in slots) has also been investigated. Further, we derive the coefficient of correlation of the lagged interdeparture intervals. Moreover, computational experiences with a variety of numerical results in the form of tables and graphs are discussed.

THE M/M/c QUEUE WITH (e, d) SETUP TIME*

The authors present a new queueing model with (e, d) setup time. Using the quasi-birth-and-death process and matrix-geometric method, the authors obtain the stationary distribution of queue length and the LST of waiting time of a customer in the system. Furthermore, the conditional stochastic decomposition results of queue length and waiting time are given.

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.

On multiserver feedback retrial queue with finite buffer

This paper deals with a multiserver feedback retrial queueing system with finite waiting position and constant retrial rate. This system is analyzed as a quasi-birth-and-death (QBD) process and the necessary and sufficient condition for stability of the system is investigated. Some important system performance measures are obtained using matrix geometric method. The effect of various parameters on the system performance measures are illustrated numerically. Finally, the algorithmic development of the full busy period for the model under consideration is discussed.

Analysis of the Discrete Time Geo/Geo/1 Queue with Single Working Vacation

Consider a discrete time Geo/Geo/1 queue with single working vacation. The server works at a lower rate rather than completely stoping service during the vacation period. Using quasi birth and death chain and matrix-geometric solution method, we obtain distributions of the number of customers in the system and the sojourn time of a customer and their stochastic decomposition structures in the stationary state. Furthermore, we derive the formulae of expected regular busy period and expected busy cycle. Finally, using some numerical examples, we analyze the effect of the parameters on the expected queue length and waiting time, and verify that our model represents some practical problems reasonably well.

Analysis of MAP/PH(1), PH(2)/2 Queue with Bernoulli Vacations

We consider a two‐heterogeneous‐server queueing system with Bernoulli vacation in which customers arrive according to a Markovian arrival process (MAP). Servers returning from vacation immediately take another vacation if no customer is waiting. Using matrix‐geometric method, the steady‐state probability of the number of customers in the system is investigated. Some important performance measures are obtained. The waiting time distribution and the mean waiting time are also discussed. Finally, some numerical illustrations are provided.

The discrete time Geom/Geom/1 queue with multiple working vacations

In this paper, we study a discrete time Geom/Geom/1 queue with multiple working vacations. Using the quasi birth and death chain and matrix-geometric solution method, we give distributions for the number of customers in system and the waiting time of a customer and their stochastic decomposition structures, and obtain distributions of the additional number of customers and additional delay. Furthermore, we derive the formulae of expected regular busy period and expected busy cycle. Finally, by numerical examples, we analyze the effect of the parameters on the expected queue length and sojourn time.

Comments on: Dynamic priority allocation via restless bandit marginal productivity indices

Queueing theory in general, and the optimal control of queues, in particular, are well developed for the case of a single server and two servers, but becomes intractable for many problems in higher dimensions. Standard methods for handling two queues, such as boundary value techniques (Cohen and Boxma 1983), or matrix geometric methods (Neuts 1981) do not extend to dimension 3 or higher. The structure of threshold and switching curve policies that characterizes optimal policies in many problems (such as admission control and routing) in dimension one and two, respectively, is usually lost for networks with more than two queues. The analysis and control of priority in queues are one of the successes of queueing theory in going beyond the frontier of dimension two, and Jose Nino-Mora has been among the leaders of these advances. Nino-Mora (2007) summarizes more than a decade of his research in that topic to which he has devoted his career since his first steps in his Ph.D. The main objectives I have chosen for this discussion note are

Queueing Analysis for GBN and SR ARQ Protocols under Dynamic Radio Link Adaptation with Non-Zero Feedback Delay

We present a queueing model for performance analysis of go-back-N (GBN) and selective repeat (SR) automatic repeat request (ARQ) protocols in wireless networks using dynamic radio link adaptation with non-instantaneous feedback. Link adaptation technique allows multi-rate transmission which is assumed to be achieved through adaptive modulation and coding. The radio link level queueing models for these two ARQ protocols are formulated in discrete time where the exact queue length and the delay statistics are obtained by using matrix geometric methods under different feedback delay values, channel and system parameters. The link layer delay statistics are useful in many ways, for example, to perform packet level admission control under statistical delay constraints. We validate the analysis by simulation and discuss useful implications of the analytical model on system performance. For dynamic link adaptation, the mode switching thresholds for the received signal-to-noise ratio (SNR) can be chosen to obtain very good link level delay performance. This SNR partitioning is shown to achieve significant cross-layer design gain compared to the case where the mode switching thresholds are chosen to maximize the physical layer throughput.

Discrete-time GI/Geo/1 queue with multiple working vacations

Consider the discrete time GI/Geo/1 queue with working vacations under EAS and LAS schemes. The server takes the original work at the lower rate rather than completely stopping during the vacation period. Using the matrix-geometric solution method, we obtain the steady-state distribution of the number of customers in the system and present the stochastic decomposition property of the queue length. Furthermore, we find and verify the closed property of conditional probability for negative binomial distributions. Using such property, we obtain the specific expression for the steady-state distribution of the waiting time and explain its two conditional stochastic decomposition structures. Finally, two special models are presented.

Distributed and Energy-Aware MAC for Differentiated Services Wireless Packet Networks: A General Queuing Analytical Framework

We present a novel queuing analytical framework for the performance evaluation of a distributed and energy-aware medium access control (MAC) protocol for wireless packet data networks with service differentiation. Specifically, we consider a node (both buffer-limited and energy-limited) in the network with two different types of traffic, namely, high-priority and low-priority traffic, and model the node as a MAP (Markovian arrival process)/PH (phase-type)/1/K nonpreemptive priority queue. The MAC layer in the node is modeled as a server and a vacation queuing model is used to model the sleep and wakeup mechanism of the server. We study standard exhaustive and number-limited exhaustive vacation models both in multiple vacation case. A setup time for the head-of-line packet in the queue is considered, which abstracts the contention and the back-off mechanism of the MAC protocol in the node. A nonideal wireless channel model is also considered, which enables us to investigate the effects of packet transmission errors on the performance behavior of the system. After obtaining the stationary distribution of the system using the matrix-geometric method, we study the performance indices, such as packet dropping probability, access delay, and queue length distribution, for high-priority packets as well as the energy saving factor at the node. Taking into account the bursty traffic arrival (modeled as MAP) and, therefore, the nonsaturation case for the queuing analysis of the MAC protocol, using phase-type distribution for both the service and the vacation processes, and combining the priority queuing model with the vacation queuing model make the analysis very general and comprehensive. Typical numerical results obtained from the analytical model are presented and validated by extensive simulations. Also, we show how the optimal MAC parameters can be obtained by using numerical optimization