Modeling parallel and distributed systems with finite workloads

In studying or designing parallel and distributed systems one should have available a robust analytical model that includes the major parameters that determine the system performance. Jackson networks have been very successful in modeling computer systems. However, the ability of Jackson networks to predict performance with system changes remains an open question, since they do not apply to systems where there are population size constraints. Also, the product-form solution of Jackson networks assumes steady-state and exponential service centers or certain specialized queueing discipline. In this paper, we present a transient model for Jackson networks that is applicable to any population size and any finite workload (no new arrivals). Using several non-exponential distributions we show to what extent the exponential distribution can be used to approximate other distributions and transient systems with finite workloads. When the number of tasks to be executed is large enough, the model approaches the product-form solution (steady-state solution). We also, study the case where the non-exponential servers have queueing (Jackson networks cannot be applied). Finally, we show how to use the model to analyze the performance of parallel and distributed systems.     

非乘积解随机Petri网的乘积形式近似求解

讨论了非乘积解随机Petri网的近似求解问题。将Marie方法引入到随机Petri网的近似分析中,利用随机Petri网中已有的结论将该方法中的分解原则推广到更一般的情形,使其应用范围更广,利用运算分析法对这些分解原则作了形式化描述,在此基础上,给出了有关结论的数学证明。最后,对这种近似方法作了误差分析,找出了产生误差的原因。实验数据表明本文所给的近似方法应用广且有效。     

A Queueing Network with Random-Delay Signals

A queueing network with an input flow of signals, along with a flow of ordinary (positive) customers, at its nodes is studied. A random time is required to activate a signal upon arrival at a node. The stationary state probability distribution for a network with single-server nodes and exponential service time distribution is derived in product form. A product-form solution for a symmetric network with Markov service at nodes is also derived.     

An approximate MVA algorithm for exponential, class-dependent multiple servers

Mean value analysis (MVA) is an efficient algorithm for determining the mean sojourn time, the mean queue length, and the throughput in a closed multiclass queueing network. It provides exact results for the class of product-form networks. Often different classes have different service requirements in FCFS queues, but such networks are not of product form. There are several possibilities to compute performance measure for such nodes and networks. In this paper we present an approximation formula for multiple-server FCFS queues with class-dependent service times as a Norton flow equivalent product node, where the departure rate of any class depends on the number of customers of all classes in the queue. We will use this approximation in the sojourn time formula of some exact and approximate MVA algorithms.     

Some Equivalence Results for Load-Independent Exponential Queueing Networks

In this paper we derive a number of results concerning the behavior of closed load-independent exponential queueing networks. It is shown that if the service rate of any station is increased (decreased), then the throughput of the network itself also increases (decreases). This is not true for product form networks in general. In addition, if the service rate at server i is increased then both the mean queue length and mean waiting time at server i decrease while both these quantities increase at all stations j ? i. The opposite effect is observed if the senrvice rate at station i is decreased. The main result of the paper is a proof of the conjective that corresponding to any general closed queueing network consisting of M stations and in which N customers circulate according to the elements of an irreducible stochastic routing matrix Q, there exists a closed load-independent exponential queueing network with the same M, N, and Q such that the mean number of customers at each station in the exponential network is equal to that in the general network. If the network throughput is specified, it is shown that this exponential network iS unique.     

Queueing analysis and optimal control of and systems

We first consider a finite-buffer single server queue where arrivals occur according to batch Markovian arrival process (BMAP). The server serves customers in batches of maximum size ‘b’ with a minimum threshold size ‘a’. The service time of each batch follows general distribution independent of each other as well as the arrival process. We obtain queue length distributions at various epochs such as, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue length, mean waiting time, probability of blocking, etc. have been obtained. Total expected cost function per unit time is also derived to determine the optimal value N^* of N at a minimum cost for given values of a and b. Secondly, we consider a finite-buffer single server queue where arrivals occur according to BMAP and service process in this case follows a non-renewal one, namely, Markovian service process (MSP). Server serves customers according to general bulk service rule as described above. We derive queue length distributions and important performance measures as above. Such queueing systems find applications in the performance analysis of communication, manufacturing and transportation systems.     

T-preemptive priority queue and its application to the analysis of an opportunistic spectrum access in cognitive radio networks

We propose a new priority discipline called the T-preemptive priority discipline. Under this discipline, during the service of a customer, at every T time units the server periodically reviews the queue states of each class with different queue-review processing times. If the server finds any customers with higher priorities than the customer being serviced during the queue-review process, then the service of the customer being serviced is preempted and the service for customers with higher priorities is started immediately. We derive the waiting-time distributions of each class in the M/G/1 priority queue with multiple classes of customers under the proposed T-preemptive priority discipline. We also present lower and upper bounds on the offered loads and the mean waiting time of each class, which hold regardless of the arrival processes and service-time distributions of lower-class customers. To demonstrate the utility of the T-preemptive priority queueing model, we take as an example an opportunistic spectrum access in cognitive radio networks, where one primary (licensed) user and multiple (unlicensed) users with distinct priorities can share a communication channel. We analyze the queueing delays of the primary and secondary users in the proposed opportunistic spectrum access model, and present numerical results of the queueing analysis.     

A unified view of product-form approximation techniques for general closed queueing networks

Two major approximate techniques have been proposed for the analysis of general closed queueing networks, namely the aggregation method and Marie's method. The idea of the aggregation technique is to replace a subsystem (a subnetwork) by a flow equivalent single-server with load-dependent service rates. The parameters of the equivalent server are obtained by analyzing the subsystem in isolation as a closed system with different populations. The idea of Marie's method is also to replace a subsystem by an equivalent exponential service station with load-dependent service rates. However, in this case, the parameters of the equivalent server are obtained by analyzing the subsystem in isolation under a load-dependent Poisson arrival process. Moreover, in Marie's case, the procedure is iterative.

In this paper we provide a general and unified view of these two methods. The contributions of this paper are the following. We first show that their common principle is to partition the network into a set of subsystems and then to define an equivalent product-form network. To each subsystem is associated a load-dependent exponential station in the equivalent network. We define a set of rules in order to partition any general closed network with various features such as general service time distributions, pupulation constraints, finite buffers, state-dependent routing. We then show that the aggregation method and Marie's method are two ways of obtaining the parameters of the equivalent network associated with a given partition. Finally, we provide a discussion pertaining to the comparison of the two methods with respect to their accuracy and computational complexity.     

A generalized method of moments for closed queueing networks

We introduce a new solution technique for closed product-form queueing networks that generalizes the Method of Moments (MoM), a recently proposed exact algorithm that is several orders of magnitude faster and memory efficient than the established Mean Value Analysis (MVA) algorithm. Compared to MVA, MoM recursively computes higher-order moments of queue lengths instead of mean values, an approach that remarkably reduces the computational costs of exact solutions, especially on models with large numbers of jobs.In this paper, we show that the MoM recursion can be generalized to include multiple recursive branches that evaluate models with different numbers of queues, a solution approach inspired by the Convolution algorithm. Combining the approaches of MoM and Convolution simplifies the evaluation of normalizing constants and leads to large computational savings with respect to the recursive structure originally proposed for MoM.     

Computational algorithms for networks of queues with rejection blocking

Summary Open, closed and mixed queueing networks with reversible routing, multiple job classes and rejection blocking are investigated. In rejection blocking networks blocking event occurs when upon completion of its service of a particular station's server, a job attempts to proceed to its next station. If, at that moment, its destination station is full, the job is rejected. The job goes back to the server of the source station and immediately receives a new service. This is repeated until the next station releases a job and a place becomes available. In the model jobs may change their class membership and general service time distributions depending on the job class are allowed. Two station types are considered: Either the scheduling discipline is symmetric, in which case the service time distributions are allowed to be general and dependent on the job class or the service time distributions at a station are all identical exponential distributions, in which case more general scheduling disciplines are allowed. An exact product form solution for equilibrium state probabilities is presented. Using the exact product form solution of the equilibrium state distribution, algorithms for computation of performance measures, such as mean number of jobs and throughputs, are derived. The complexity of the algorithms is discussed.     

On the nonconcavity of throughput in certain closed queueing networks

Analytic queueing network models are being used to analyze various optimization problems such as server allocation, design and capacity issues, optimal routing, and workload allocation. The mathematical properties of the relevant performance measures, such as throughput, are important for optimization purposes and for insight into system performance.We show that for closed queueing networks of m arbitrarily connected single server queues with n customers, throughput, as a function of a scaled, constrained workload, is not concave. In fact, the function appears to be strictly quasiconcave. There is a constraint on the total workload that must be allocated among the servers in the network. However, for closed networks of two single server queues, we prove that our scaled throughput is concave when there are two customers in the network and strictly quasi-concave when there are more than two customers. The mathematical properties of both the scaled throughput and reciprocal throughput are demonstrated graphically for closed networks of two and three single server queues.     

Analytical and simulation modeling of a multi-server queue with Markovian arrivals and priority services

We consider a multi-server queueing system in which two types of customers arrive according to a Markovian arrival process. Type 1 customers have preemptive priority over Type 2 customers. A Type 2 arrival finding all servers busy will be lost. However, a Type 1 customer finding all servers busy with at least one Type 2 in service will get into service by pre-empting one of the Type 2 customers in service. Pre-empted Type 2 customers enter into a buffer of finite capacity. These (preempted) customers eventually leave the system after completing a service. In the case of exponential services, this model is studied analytically in steady-state by exploiting the special nature of the queueing model. A number of useful performance measures along with some illustrative examples are reported. In the case of non-exponential services, we simulate the model and discuss the effect of the variatio the services on some selected performance measures.     

The stationary distribution invariance of states in a closed queueing network with temporarily non-active customers

A stationary functioning of a closed queueing network with temporarily non-active customers is analyzed. Non-active customers are located at network nodes in queues, being not serviced. For a customer, the feasibility of passing from its ordinary state to the temporarily non-active state (and backwards) is provided. Service times of customers at different nodes possess arbitrary distributions. Finally, the stationary distribution invariance of network states is established with respect to the functional form of customer service time distributions under fixed first-order moments.     

MRE hierarchical decomposition of general queueing network models

Summary The principle of Minimum Relative Entropy (MRE), given fully decomposable subset and aggregate mean queue length, utilisation and flow-balance constraints, is used in conjunction with asymptotic connections to infinite capacity queues, to derive new analytic approximations for the conditional and marginal state probabilities of single class general closed queueing network models (QNMs) in the context of a multilevel variable aggregation scheme. The concept of subparallelism is applied to preserve the flow conservation and a universal MRE hierarchical decomposition algorithm is proposed for the approximate analysis of arbitrary closed queueing networks with single server queues and general service-times. Heuristic criteria towards an optimal coupling of network's units at each level of aggregation are suggested. As an illustration, the MRE algorithm is implemented iteratively by using the Generalised Exponential (GE) distributional model to approximate the service and asymptotic flow processes in the network. This algorithm captures the exact solution of separable queueing networks, while for general queueing networks it compares favourably against exact solutions and known approximations.This work is sponsored by the Science and Engineering Research Council (SERC), UK, under grant GR/F29271     

Parameter estimation for product-form distributions of queueing networks

Basic parameters of a queueing network are its routing matrix, arrival flow rate, and service rates at network nodes. To estimate these parameters, one has to solve a system of balance equations. In turn, a product-form limiting distribution of the number of customers at the network nodes is defined through loading factors. Therefore, in the paper we propose to estimate loading factors through estimates of the limiting distribution based on observations of the number of customers at the nodes. This makes it possible to avoid solving a system of balance equations. This algorithm is realized for Jackson networks: classical, in a random environment, with blocked transitions.     

A discrete-time retrial queue with negative customers and unreliable server

This paper treats a discrete-time single-server retrial queue with geometrical arrivals of both positive and negative customers in which the server is subject to breakdowns and repairs. Positive customers who find sever busy or down are obliged to leave the service area and join the retrial orbit. They request service again after some random time. If the server is found idle or busy, the arrival of a negative customer will break the server down and simultaneously kill the positive customer under service if any. But the negative customer has no effect on the system if the server is down. The failed server is sent to repair immediately and after repair it is assumed as good as new. We analyze the Markov chain underlying the queueing system and obtain its ergodicity condition. The generating functions of the number of customers in the orbit and in the system are also obtained along with the marginal distributions of the orbit size when the server is idle, busy or down. Finally, some numerical examples show the influence of the parameters on some crucial performance characteristics of the system.     

Robust design of server capability in M/M/1 queues with both partly random arrival and service rates

In this paper we consider a robust design of controllable factors related to the server capability in M/M/1 queues where both arrival and service rates are assumed to be partly random. The performance of an individual queue is measured in terms of the random traffic intensity parameter defined as the ratio of the arrival rate to the service rate where both rates are functions of associated characteristics of an individual queue and a random error. We utilize the empirical Bayes estimator of the traffic intensity parameter and employ a Monte-Carlo simulation to find the optimal levels of server characteristics with respect to mean squared error. An example is given to illustrate how the proposed procedures can be applied to the robust design of a transmission line.Scope and purposeRobust design is an important issue and has been extensively applied to both product and manufacturing process design so that the resulting quality can consistently satisfy customers under the variation of some uncontrollable factors. We apply this concept to design server capability in a queueing system for given arrival rates. In order to reflect random phenomena, we use Bayesian approach to estimate parameters in the given queueing model. We expect that the resulting robust design procedure can be effectively utilized for budgeting server levels of various queueing systems.     

Markovian bulk service queue with delayed vacations

An M/M(a, b)/1 queueing system with multiple vacations is studied, in which if the number of customers in the queue is a - 1 either at a service completion epoch or at a vacation completion point, the server will wait for an exponential time in the system which is called the changeover time. During this changeover time if there is an arrival the server will start service immediately, otherwise at the end of the changeover time the server will go for a vacation. The duration of vacation is also exponential. This paper is concerned with the determination of the stationary distribution of the number of customers in the queue and the waiting time distribution of an arriving customer. The expected queue length is also obtained. Sample numerical illustrations are given.     

The concert queueing game: to wait or to be late

We introduce the concert (or cafeteria) queueing problem: A finite but large number of customers arrive into a queueing system that starts service at a specified opening time. Each customer is free to choose her arrival time (before or after opening time), and is interested in early service completion with minimal wait. These goals are captured by a cost function which is additive and linear in the waiting time and service completion time, with coefficients that may be class dependent. We consider a fluid model of this system, which is motivated as the fluid-scale limit of the stochastic system. In the fluid setting, we explicitly identify the unique Nash-equilibrium arrival profile for each class of customers. Our structural results imply that, in equilibrium, the arrival rate is increasing up until the closing time where all customers are served. Furthermore, the waiting queue is maximal at the opening time, and monotonically decreases thereafter. In the simple single class setting, we show that the price of anarchy (PoA, the efficiency loss relative to the socially optimal solution) is exactly two, while in the multi-class setting we develop tight upper and lower bounds on the PoA. In addition, we consider several mechanisms that may be used to reduce the PoA. The proposed model may explain queueing phenomena in diverse settings that involve a pre-assigned opening time.     

Analytical observations for a multiservice node

Motivated by a desire to understand a GI/∞ node in a closed queueing network environment, we study in this paper a finite population multiserver model, where the number of processes is equal to the population in the network. We study its steady-state behavior under various scenarios, e.g., the first three moments of the arrival and service distributions. We show that when the arrivals are bursty, an increase in service time variation can actually decrease congestion. Furthermore, such a decrease is also observed for an increase in the skewness in the arrivals. Additionally, several insensitivity properties in limiting behaviors are observed. In particular, the finite population GI/GI///s-loop is insensitive to the service time distribution when the arrival distribution either has an infinite variance (a form of heavy tail), or is highly skewed. In the latter case, the congestion approaches that of the all Markovian system M/M///s-loop. We conclude that the influence of the third moment can be very significant, indicating that the first two moments do not suffice to characterize the performance measures under consideration.