Exponential Service Time Distribution Research Articles

Response time analysis of parallel computer and storage systems

Fork-join structures have gained increased importance in recent years as a means of modeling parallelism in computer and storage systems. The basic fork-join model is one in which a job arriving at a parallel system splits into K independent tasks that are assigned to K unique, homogeneous servers. In the paper, a simple response time approximation is derived for parallel systems with exponential service time distributions. The approximation holds for networks modeling several devices, both parallel and nonparallel. (In the case of closed networks containing a stand-alone parallel system, a mean response time bound is derived.) In addition, the response time approximation is extended to cover the more realistic case wherein a job splits into an arbitrary number of tasks upon arrival at a parallel system. Simulation results for closed networks with stand-alone parallel subsystems and exponential service time distributions indicate that the response time approximation is, on average, within 3 percent of the seeded response times. Similarly, simulation results with nonexponential distributions also indicate that the response time approximation is close to the seeded values. Potential applications of our results include the modeling of data placement in disk arrays and the execution of parallel programs in multiprocessor and distributed systems.

The MMCPP/GE/ c Queue

We obtain the queue length probability distribution at equilibrium for a multi-server, single queue with generalised exponential (GE) service time distribution and a Markov modulated compound Poisson arrival process (MMCPP) – i.e., a Poisson point process with bulk arrivals having geometrically distributed batch size whose parameters are modulated by a Markovian arrival phase process. This arrival process has been considered appropriate in ATM networks and the GE service times provide greater flexibility than the more conventionally assumed exponential distribution. The result is exact and is derived, for both infinite and finite capacity queues, using the method of spectral expansion applied to the two dimensional (queue length by phase of the arrival process) Markov process that describes the dynamics of the system. The Laplace transform of the interdeparture time probability density function is then obtained. The analysis therefore could provide the basis of a building block for modelling networks of switching nodes in terms of their internal arrival processes, which may be both correlated and bursty.

Analysis of a shared-memory multiprocessor via a novel queuing model

The study on the delay effects caused by memory interference in a parallel processing system with shared memory was implemented using a Machine Repair queuing Model (MRM). In our model the Repair Server has a constant service time and the parallel operating Job Servers have negative exponential service time distributions. Mean Value Analysis (MVA) was used to solve this particular MRM. However, to apply MVA successfully in this model with heterogeneous servers, the response time equation of the Repair Server has been modified. This modified equation was derived by combining mathematics and heuristics. To check this novel method on its accuracy, comparisons were made with as many as 1200 simulations, having a wide spread of model parameters. The results are extremely good, differences between the analytical results and the simulations are always smaller than 2%.

Probabilistic load scheduling with priorities in distributed computing systems

Scheduling of processing loads play an important role in the performance of a distributed computing system/network. This is more important when the distributed computing system consists of many heterogeneous processors, with different processing capabilities. The scheduling strategies should also take into account the priorities given to the loads, when there is more than one type of jobs utilizing the system. A methodology is presented to obtain optimal load scheduling strategies, when there are two type of loads in a heterogeneous distributed computing system. In this paper a distributed computing system/network in which two type of loads (or jobs) namely local loads and global loads arrive is considered. The local loads originate at the individual processors/nodes in the network and have to be processed at the same processors. The global loads originate at the central scheduler and have to be distributed among the processors in the system. The individual processors in the system assign different priorities to local and global loads. A priority queuing optimization model of this situation is formulated as a nonlinear programming problem and a solution methodology is presented. The arrival rate of the global and local loads are assumed to follow Poisson distribution and the service time of the global and local loads at the processors follow a general or exponential service time distribution. Both the situations are analyzed. The objective of this study is to minimize the response time of the global loads in the network.

Performance evaluation of open queueing networks with arbitrary configuration and finite buffers

We consider an open queueing network consisting of servers linked in an arbitrary configuration with exponential service times and separated by intermediate finite buffers. The model allows any number of saturated servers (never starved) and exit servers (never blocked). The considered blocking mechanism is blocking-after-service. In the case of simultaneous blocking, blocked customers enter the destination node on a first-blocked-first-enter basis. If feedback loops exist in the network, deadlocks may arise. We assume that a deadlock is detected and resolved instantaneously by transferring all the blocked customers simultaneously. In this paper, we present an approximation method to analyze this kind of networks. The method decomposes the original network into subsystems. Each subsystem is composed of one or many upstream servers and one downstream server, separated by a finite buffer. The upstream and downstream servers are characterized by exponential service time distributions. Based on the symmetrical approach [2], we develop an algorithm to determine the unknown parameters corresponding to each subsystem. The class of models considered in this paper includes the class of open loss models for feed-forward networks considered by Lee and Pollock [11]. For this class of models, we can show that the system of equations in our algorithm is equivalent to the one used in the algorithm proposed by Lee and Pollock. As a result, our algorithm provides the same results as the algorithm of Lee and Pollock for this class of models. However, it is observed that our algorithm takes less CPU execution time than the one proposed by Lee and Pollock. For the cases of networks with feedback loops, extensive numerical experiments show that the new algorithm, in general, converges very fast and yields accurate results compared with those obtained by simulation as long as deadlocks do not occur too frequently. Moreover, for the merge configuration, we provide the proof of the convergence of the algorithm as well as the existence and uniqueness of the solution by exploiting the properties associated with a symmetric approach.

Arbitrary open queueing networks with server vacation periods and blocking

A cost-effective decomposition algorithm is proposed for the approximate analysis of arbitrary first-come-first-served (FCFS) open queueing network models (QNMs) with Generalised Exponential (GE) external interarrival time and service time distributions, multiple vacation periods, exhaustive service scheme and repetitive service blocking of random destination (RS-RD). The algorithm is based on a new product-form approximation suggested by the information theoretic principle of Minimum Relative Entropy (MRE), subject to marginal mean value constraints and a prior joint distribution estimate of the corresponding open QNM with infinite capacity. The censored GEyGEy1yN queue with exponential server vacations is solved via relative entropy minimisation and in conjunction with associated effective flow streams (departure, splitting, merging), plays the role of an efficient building block in the solution process. An exhaustive validation study against simulation is included to illustrate the credibility of the MRE results.

On First-Come First-Served Versus Random Service Discipline in Multiclass Closed Queueing Networks

We consider multiclass closed queueing networks. For these networks, a lot of work has been devoted to characterizing and weakening the conditions under which a product-form solution is obtained for the steady-state distribution. From this work, it is known that, under certain conditions, all networks in which each of the stations has either the first-come first-served or the random service discipline lead to the same (product-form expressions for the) steady-state probabilities of the (aggregated) states that for each station and each job class denote the number of jobs in service and the number of jobs in the queue. As a consequence, all these situations also lead to the same throughputs for the different job classes. One of the conditions under which these equivalence results hold states that at each station all job classes must have the same exponential service time distribution. In this paper, it is shown that these equivalence results can be extended to the case with different exponential service times for jobs of different classes, if the network consists of only one single-server or multiserver station. This extension can be made despite of the fact that the network is not a product-form network anymore in that case. The proof is based on the reversibility of the Markov process that is obtained under the random service discipline. By means of a counterexample, it is shown that the extension cannot be made for closed network with two or more stations.

A Transform-Free Approximation for the Finite Capacity M/G/s Queue

This paper develops a transform-free approximation for the steady-state queue-length distribution in an M/G/s queue with finite waiting spaces. The approximation is obtained by using a conservation law and some heuristics. It is shown that the approximation is exact for the cases with either no extra waiting space, exponential service-time distribution, or a certain two-parameter family of service-time distributions. It is also shown that the approximation has the same light-traffic properties as the known light-traffic limit theorem for the infinite capacity case when the number of waiting spaces is not less than one.

Analysis of production control systems kanban and CONWIP

In this paper we describe and classify different pull production systems. The production control systems kanban and CONWIP are then analysed with respect to production rate and average W1P. We examine single product flow lines with exponential service time distributions and unlimited demand at the final buffer (saturated lines). We show that the distribution of cards (kanbans) has a significant effect on the performance of kanban systems. Different types of kanban control mechanisms show equivalent performance data, if the distribution pattern is adapted accordingly. Our research shows that the kanban system is more flexible with respect to a given objective than the CONWIP system, because in addition to the total card number the card distribution is a control parameter. Moreover for a given production rate the average WIP is lower in a kanban system than in a CONWIP system. We identify the average WIP in the interstage buffers as important parameter and describe the ‘WIP bowl phenomenon’ as result of optimum card distributions. Concluding remarks and directions for future research conclude the paper.

Analysis of balanced fork-join queueing networks

This paper presents an analysis of closed, balanced, fork-join queueing networks with exponential service time distributions. The fork-join queue is mapped onto two non-parallel networks, namely, a serial-join model and a state-dependent model. Using these models, it is proven that the proportion of the number of jobs in the different subsystems of the fork-join queueing network remains constant, irrespective of the multiprogramming level. This property of balanced fork-join networks is used to compute quick, inexpensive bounds for arbitrary fork-join networks.

PERFORMANCE ANALYSIS OF HETEROGENEOUS TRAFFIC ON AN INTEGRATED NETWORK LINK WITH FINITE WAITING ROOM AND ANTICIPATED-RELEASE RESERVATION POLICY

We consider a system where the superposition of two heterogeneous Poisson traffic streams is offered to an integrated network link in synchronous transfer mode, where one stream follows the blocked-and-cleared mode (‘loss’ mode) and the other can wait (finitely) if bandwidth is not available for connection at the time of arrival (‘hold’ mode). We assume that each stream has different bandwidth requirements per call. A reservation scheme, called anticipated-release policy, is introduced where an arrival is accepted into a waiting room only if the amount of time this customer is expected to wait is within acceptable limits. For such a loss/hold system, we provide analytical performance models for exponential service time distributions for both streams as well as for the non-exponential service time distribution case for the traffic stream in ‘hold’ mode. We also present a method on how to model the waiting time distribution of the traffic stream with ‘hold’ mode. From numerical studies, we observe that blocking can be reduced considerably for both services just by introducing a small waiting room for one traffic class compared to ‘loss’ mode for both traffic classes. Furthermore, this holds true for the case when a maximum tolerable time limit is imposed on the waiting. Finally, our results indicate that this loss/hold scenario with limited waiting room appears to be virtually insensitive to the service time distribution of the ‘hold’ mode traffic.

Asymptotic analysis of the effect of arrival model uncertainties in some optimal routing problems

We study the effect of arrival model uncertainties on the optimal routing in a system of parallel queues. For exponential service time distributions and Bernoulli routing, the optimal mean system delay generally depends on the interarrival time distribution. Any error in modeling the arriving process will cause a model-based optimal routing algorithm to produce a mean system delay higher than the true optimum. In this paper, we present an asymptotic analysis of the behavior of this error under heavy traffic conditions for a general renewal arrival process. An asymptotic analysis of the error in optimal mean delay due to uncertainties in the service time distribution for Poisson arrivals was reported in Ref. 6, where it was shown that, when the first moment of the service time distribution is known, this error in performance vanishes asymptotically as the traffic load approaches the system capacity. In contrast, this paper establishes the somewhat surprising result that, when only the first moment of the arrival distribution is known, the error in optimal mean delay due to uncertainties in the arrival model is unbounded as the traffic approaches the system capacity. However, when both first and second moments are known, the error vanishes asymptotically. Numerical examples corroborating the theoretical results are also presented.

PLANNING TIMELY ARRIVALS TO A STOCHASTIC PRODUCTION OR SERVICE SYSTEM

A stochastic planning problem of determining the optimal arrival times for N customers, each to be assigned to one of K equal time slots, is considered. The objective is to minimize the total system cost, which is composed of the customer waiting cost and the server availability cost. This optimal arrival schedule is examined for a single server system with either exponential or Erlang-fc service time distribution. Two versions of this planning arrival problem are considered, a dynamic version and a static version. The dynamic problem requires planning decisions to be made at the beginning of each time slot, while in the static problem all decisions must be made at the beginning of the first time slot. The dynamic problem is solved by dynamic programming. The structure of the optimal dynamic policy is identified and used to solve the dynamic problem efficiendy. A branch-and-bound algorithm, which uses the solution to the dynamic problem, is developed to solve the static problem. The results can be used to...

Approximate decomposition for closed queueing network models of FMSs with a block-and-wait and state-dependent routing mechanism

Closed queueing networks with a central server configuration are used to analyse the performance of Flexible Manufacturing Systems. The blocking phenomenon of cart-type Material Handling Systems (MHS) due to the finite capacity of local buffers is addressed and a state-dependent routing scheme is introduced to assure that the MHS does not send parts to workstations where blocking events have already occurred. Under the assumptions of exponential service time distribution and the FCFS discipline at workstations, an efficient approximate decomposition approach is developed to evaluate the performance measures of the resulting non product-form queueing networks with a block-and-wait mechanism. The accuracy of the proposed method is established by comparison with the results of simulation experiments.

A nonpreemptive priority multiserver queueing system with general bulk service and heterogeneous arrivals

In this paper a multiserver nonpreemptive priority queueing system consisting of two types of customer units with Poisson arrival and exponential service time distribution is analysed. The lower priority units are served in batches according to a general bulk service rule. The higher priority units have nonpreemptive priority over lower priority units. The higher priority units are served singly. The steady state probability vectors of the number of customers in the queue are obtained using the modified matrix-geometric method. The stability condition and mean queue length of customers are obtained. Numerical results are also presented.

A preemptive priority queue with general bulk service and heterogeneous arrivals

A single-server preemptive priority queueing system consisting of two types of units, with unlimited Poisson input of high priority and limited Poisson input of low priority, and exponential service time distribution is studied. The higher-priority units are served in batches according to a general bulk service rule and they have preemptive priority over lower-priority units. The server will stop servicing a low-priority unit if the size of the queue of high-priority units reaches the minimum number of units required for the bulk service. The server will start servicing a low-priority unit if the size of the high-priority input queue is less than the minimum required for the general bulk service. The probabilities of the number of customers in the queue in steady state and the stability condition are obtained using the matrix-geometric algorithmic approach.

Monotonicity results for queues with doubly stochastic Poisson arrivals: Ross's conjecture

In this paper, we compare queueing systems that differ only in their arrival processes, which are special forms of doubly stochastic Poisson (DSP) processes. We define a special form of stochastic dominance for DSP processes which is based on the well-known variability or convex ordering for random variables. For two DSP processes that satisfy our comparability condition in such a way that the first process is more ‘regular' than the second process, we show the following three results: (i) If the two systems are DSP/GI/1 queues, then for all f increasing convex, with V (i), i = 1 and 2, representing the workload (virtual waiting time) in system. (ii) If the two systems are DSP/M(k)/1→ /M(k)/l ∞ ·· ·∞ /M(k)/1 tandem systems, with M(k) representing an exponential service time distribution with a rate that is increasing concave in the number of customers, k, present at the station, then for all f increasing convex, with Q (i), i = 1 and 2, being the total number of customers in the two systems. (iii) If the two systems are DSP/M(k)/1/N systems, with N being the size of the buffer, then where denotes the blocking (loss) probability of the two systems. A model considered before by Ross (1978) satisfies our comparability condition; a conjecture stated by him is shown to be true.

Monotonicity results for queues with doubly stochastic Poisson arrivals: Ross's conjecture

In this paper, we compare queueing systems that differ only in their arrival processes, which are special forms of doubly stochastic Poisson (DSP) processes. We define a special form of stochastic dominance for DSP processes which is based on the well-known variability or convex ordering for random variables. For two DSP processes that satisfy our comparability condition in such a way that the first process is more ‘regular' than the second process, we show the following three results: (i) If the two systems are DSP/GI/1 queues, then for all f increasing convex, with V(i), i = 1 and 2, representing the workload (virtual waiting time) in system. (ii) If the two systems are DSP/M(k)/1→ /M(k)/l ∞ ·· ·∞ /M(k)/1 tandem systems, with M(k) representing an exponential service time distribution with a rate that is increasing concave in the number of customers, k, present at the station, then for all f increasing convex, with Q(i), i = 1 and 2, being the total number of customers in the two systems. (iii) If the two systems are DSP/M(k)/1/N systems, with N being the size of the buffer, then where denotes the blocking (loss) probability of the two systems. A model considered before by Ross (1978) satisfies our comparability condition; a conjecture stated by him is shown to be true.

A model of channel allocation in multichannel local networks

We consider a queueing model with preferentially ordered parallel servers that models a demand assignment channel allocation scheme used in multichannel local networks. We devise a method for reducing the number of system states to construct a tractable model from which we derive, for each n, the probability distribution that k of the first n ordered servers are busy. From these distributions, utilizations of individual servers are derived. Then assuming Poisson arrivals and exponential service time distributions, we develop analytic expressions for server utilizations. For the finite population model, where analytic expressions cannot be derived, we develop an efficient polynomial algorithm to compute server utilizations. Simple expressions for calculating upper bounds on finite population model server utilizations are also derived.

Approximate MVA for closed queueing network models of FMS with a block-and-wait mechanism

Closed queueing networks with a central server configuration are used to model Flexible Manufacturing Systems (FMS) in which the material handling system consists of a single cart and the finite capacity of local buffers leads to blocking. Under the assumptions of an exponential service time distribution and a block-and-wait mechanism, an efficient approximate Mean Value Analysis (MVA) method is developed to evaluate the performance measures of the resulting non product-form queueing networks. The accuracy of the proposed method is established by comparison with the results of simulation experiments.