Infinite Server Queue Research Articles

Tandem Queues with Impatient Customers for Blood Screening Procedures

We study a blood testing procedure for detecting viruses like HIV, HBV and HCV. In this procedure, blood samples go through two screening steps. The first test is ELISA (antibody Enzyme Linked Immuno-Sorbent Assay). The portions of blood which are found not contaminated in this first phase are tested in groups through PCR (Polymerase Chain Reaction). The ELISA test is less sensitive than the PCR test and the PCR tests are considerably more expensive. We model the two test phases of blood samples as services in two queues in series; service in the second queue is in batches, as PCR tests are done in groups. The fact that blood can only be used for transfusions until a certain expiration date leads, in the tandem queue, to the feature of customer impatience. Since the first queue basically is an infinite server queue, we mainly focus on the second queue, which in its most general form is an S-server M/G [k, K]/S + G queue, with batches of sizes which are bounded by k and K. Our objective is to maximize the expected profit of the system, which is composed of the amount earned for items which pass the test (and before their patience runs out), minus costs. This is done by an appropriate choice of the decision variables, namely, the batch sizes and the number of servers at the second service station. As will be seen, even the simplest version of the batch queue, the M/M [k, K]/1 + M queue, already gives rise to serious analytical complications for any batch size larger than 1. These complications are discussed in detail, and handled for K = 2. In view of the fact that we aim to solve realistic optimization problems for blood screening procedures, these analytical complications force us to take recourse to either a numerical approach or approximations. We present a numerical solution for the queue length distribution in the M/M [k, K]/S + M queue and then formulate and solve several optimization problems. The power-series algorithm, which is a numerical-analytic method, is also discussed.

A Hybrid Queueing Model With Imperfect Debugging For Component Software Reliability Analysis

Abstract With the growing size and complexity of software applications, research in the area of component software reliability analysis has gained prominence. To ensure analytical tractability, traditional approaches usually ignore fault correction process of component software based on simplifying assumptions of instantaneous and perfect debugging. As a result, the estimates obtained from these models tend to be optimistic. To obtain realistic estimates, it is desirable that the assumptions of instantaneous and perfect debugging be amended. In this paper we discuss fault repair policy according to which debugging to various components may be conducted in integration testing of component software. We then propose a hybrid infinite server queueing model to describe fault correction process with consideration of the possibility of imperfect debugging. Based on fault detection process and repair policy, the model can be resolved to represent component software reliability growth during integration testing. F...

Performability Analysis for Software-Intensive System Considering Variety of Tasks and Operation-Oriented Restoration

In this paper, we discuss software performability evaluation considering the real-time property. The time-dependent behavior of the system alternating between up and down state is described by the Markov process. Then we incorporate the operation-oriented restoration scenario into the model, i.e., we consider the following two types of restoration: one is the restoration with debugging and the other is without debugging. Assuming that the software system can process the multiple tasks simultaneously, we also consider the variety of tasks in terms of the task arrival process and the processing time limit. We describe the arrival process of the tasks follows a nonhomogeneous Poisson process and treat the processing time limit as a random variable. We analyze the distribution of the number of tasks whose processes can be completed within the processing time limit with the infinite server queueing model. From the model, we derive several software performability measures considering the real-time property. Finally, we illustrate several numerical examples of the measures to investigate the impact of the variety of tasks and the operational restoration on the system performability evaluation.

Semi-Markov-Modulated Infinite-Server Queues: Approximations by Time-Scaling

This article studies an infinite-server queue in a semi-Markov environment: the queue's input rate is modulated by a semi-Markovian background process, and the service times are assumed to be exponentially distributed. The primary objective of this article is to propose approximations for the queue-length distribution, based on time-scaling arguments. The analysis starts with an explicit analysis of the cases in which the transition times of the modulating semi-Markov process are either all deterministic or all exponential. We use these results to obtain approximations under time-scalings; both a quasi-stationary regime (in which time is slowed down) and a fluid-scaling regime (in which time is sped up) are considered. Notably, in the latter regime, the limiting distribution of the number of customers present is Poisson, irrespective of the distribution of the transition times. The accuracy of the resulting approximations is illustrated by several numerical experiments, that moreover give an indication of the speed of convergence in both regimes, for various distributions of the transition times. The last section derives conditions under which the distribution of the number of customers present is Poisson (in an exact sense, i.e., not in a limiting regime).

Stochastic performability measurement for software system with random performance degradation and field-oriented restoration

This paper constructs the integrated software performability model reflecting the actual field environment. The time-dependent behavior of the system alternating between up and down states is described by the Markov process. Then we incorporate the undesirable operational state and the field-oriented restoration scenario into the model, i.e., we consider the following two types of operational state: one is operating with the desirable performance level according to specification and the other is with degraded performance level, and the following two types of restoration: one is the restoration with debugging and the other is without debugging. Assuming that the software system can process the multiple tasks simultaneously, we describe the arrival process of the tasks follows a nonhomogeneous Poisson process and treat the processing time limit imposed on each task as a random variable. We analyze the distribution of the number of tasks whose processes can be completed within the processing time limit with the infinite server queueing model. From the model, we derive several software performability measures considering the real-time property. Finally, we illustrate several numerical examples of the measures to investigate the impacts of the difference of the operational performance levels and the field-oriented restoration on the system performability evaluation.

Software reliability analysis and assessment using queueing models with multiple change-points

Over the past three decades, many software reliability growth models (SRGMs) have been proposed, and they can be used to predict and estimate software reliability. One common assumption of these conventional SRGMs is to assume that detected faults will be removed immediately. In reality, this assumption may not be reasonable and may not always occur. During debugging, developers need time to reproduce the failure, identify the root causes of faults, fix them, and then re-run the software. From some experiments or observations, the fault correction rate may not be a constant and could be changed at certain points as time proceeds. Consequently, in this paper, we will investigate and study how to apply queueing models to describe the fault detection and correction processes during software development. We propose an extended infinite server queueing model with multiple change-points to predict and assess software reliability. Experimental results based on real failure data show that the proposed model can depict the change of fault correction rates and predict the behavior of software development more accurately than traditional SRGMs.

Two-parameter heavy-traffic limits for infinite-server queues

In order to obtain Markov heavy-traffic approximations for infinite-server queues with general non-exponential service-time distributions and general arrival processes, possibly with time-varying arrival rates, we establish heavy-traffic limits for two-parameter stochastic processes. We consider the random variables Q e (t,y) and Q r (t,y) representing the number of customers in the system at time t that have elapsed service times less than or equal to time y, or residual service times strictly greater than y. We also consider W r (t,y) representing the total amount of work in service time remaining to be done at time t+y for customers in the system at time t. The two-parameter stochastic-process limits in the space D([0,∞),D) of D-valued functions in D draw on, and extend, previous heavy-traffic limits by Glynn and Whitt (Adv. Appl. Probab. 23, 188–209, 1991), where the case of discrete service-time distributions was treated, and Krichagina and Puhalskii (Queueing Syst. 25, 235–280, 1997), where it was shown that the variability of service times is captured by the Kiefer process with second argument set equal to the service-time c.d.f.

Transient Solution to an infinite Server Queue with Varying Arrival and Departure Rate

Problem statement: In many potential application of queueing theory, the transient solution of queueing system is important. Approach: This study presented the transient solution for infinite server queues with Poisson arrivals and exponential service times when the parameters of both distributions are allowed to vary with time. Based on generating functions technique which results in a simple differential equation. Using the properties of Bessel functions in the solution of this differential equation, the solution of an infinite server queues can be given in simple form. Results: The researcher obtained the transient solution an infinite server queues with Poisson arrivals and exponential service times when the parameters of both distributions are allowed to vary with time and prove that some past results are special case from his results. Conclusion: These results indicated that the probabilities can be extracted in a direct way.

Importance sampling algorithms for first passage time probabilities in the infinite server queue

Abstract This paper applies importance sampling simulation for estimating rare event probabilities of the first passage time in the infinite server queue with renewal arrivals and general service time distributions. We consider importance sampling algorithms which are based on large deviations results of the infinite server queue, and we consider an algorithm based on the cross-entropy method, where we allow light-tailed and heavy-tailed distributions for the interarrival times and the service times. Efficiency of the algorithms is discussed by simulation experiments.

Time-Dependent Analysis of Some Infinite Server Queues with Bulk Poisson Arrivals

Numerical computational formulae for the transient distribution of the number of customers in system in the infinite server queue with bulk Poisson arrivals are derived using generating function techniques. Equilibrium analysis follows as a special case. Various bulk size and service time distributions are considered, and numerical examples illustrate the use of the approach.

Measuring statistical heterogeneity: The Pietra index

There are various ways of quantifying the statistical heterogeneity of a given probability law: Statistics uses variance — which measures the law’s dispersion around its mean; Physics and Information Theory use entropy — which measures the law’s randomness; Economics uses the Gini index — which measures the law’s egalitarianism. In this research we explore an alternative to the Gini index–the Pietra index–which is a counterpart of the Kolmogorov–Smirnov statistic. The Pietra index is shown to be a natural and elemental measure of statistical heterogeneity, which is especially useful in the case of asymmetric and skewed probability laws, and in the case of asymptotically Paretian laws with finite mean and infinite variance. Moreover, the Pietra index is shown to have immediate and fundamental interpretations within the following applications: renewal processes and continuous time random walks; infinite-server queueing systems and shot noise processes; financial derivatives. The interpretation of the Pietra index within the context of financial derivatives implies that derivative markets, in effect, use the Pietra index as their benchmark measure of statistical heterogeneity.

DEBUGGING PROCESS-ORIENTED DISCRETE SOFTWARE RELIABILITY MODELING

This paper discusses an new approach for discrete-time software reliability growth modeling based on an discrete-time infinite server queueing model, which describes a debugging process in a testing phase. Our approach enables us to develop discrete-time software reliability growth models (SRGMs) which could not be developed under conventional discrete-time modeling approaches. This paper also discuss goodness-of-fit comparisons of our discrete-time SRGMs with conventional continuous-time SRGMs in terms of the criterion of the mean squared errors, and show numerical examples for software reliability analysis of our models by using actual data.

Time-dependent analysis for refused admissions in clinical wards

For capacity planning issues in health care, such as the allocation of hospital beds, the admissions rate of patients is commonly assumed to be constant over time. In addition to the purely random fluctuations, there is also typically a predictable pattern in the number of arriving patients. For example, roughly 2/3 of the admitted patients at an Intensive Care Unit arrives during office hours. Also, most of the scheduled admissions occur during weekdays instead of during the weekend. Using approximations based on the infinite-server queue, we analyze an M t /H/s/s model to determine the impact of the time-dependent arrival pattern on the required number of operational beds and fraction of refused admissions for clinical wards. In particular, the results show that the effect of the daily pattern is rather limited for clinical wards in contrast to the week-weekend pattern, for which the difference in the fraction of refused admissions across the week is considerable. We also show that an increased variability in length of stay distribution has a stabilizing effect on the time-dependent required number of beds. Finally, we demonstrate a method to determine the required number of beds across the week.

Heavy-traffic limits for many-server queues with service interruptions

We establish many-server heavy-traffic limits for G/M/n+M queueing models, allowing customer abandonment (the +M), subject to exogenous regenerative service interruptions. With unscaled service interruption times, we obtain a FWLLN for the queue-length process, where the limit is an ordinary differential equation in a two-state random environment. With asymptotically negligible service interruptions, we obtain a FCLT for the queue-length process, where the limit is characterized as the pathwise unique solution to a stochastic integral equation with jumps. When the arrivals are renewal and the interruption cycle time is exponential, the limit is a Markov process, being a jump-diffusion process in the QED regime and an O---U process driven by a Levy process in the ED regime (and for infinite-server queues). A stochastic-decomposition property of the steady-state distribution of the limit process in the ED regime (and for infinite-server queues) is obtained.

Performability Modeling for Software System with Performance Degradation and Reliability Growth

In this paper, we discuss software performability evaluation considering the real-time property; this is defined as the attribute that the system can complete the task within the stipulated response time limit. We assume that the software system has two operational states from the viewpoint of the end users: one is operating with the desirable performance level according to specification and the other is with degraded performance level. The dynamic software reliability growth process with performance degradation is described by the extended Markovian software reliability model with imperfect debugging. Assuming that the software system can process the multiple tasks simultaneously and that the arrival process of the tasks follows a nonhomogeneous Poisson process, we analyze the distribution of the number of tasks whose processes can be completed within the processing time limit with the infinite server queueing model. We derive several software performability measures considering the real-time property; these are given as the functions of time and the number of debugging activities. Finally, we illustrate several numerical examples of the measures to investigate the impact of consideration of the performance degradation on the system performability evaluation.

Performance Evaluation of Software Task Processing Based on Markovian Perfect Debugging Model

This paper proposes a new model by combining an infinite-server queueing model for multi-task processing software system with a perfect debugging model based on Markov process with two types of faults suggested by Lee et al. (2001). We apply this model for module and integration testing in the testing process. Also, we compute several measure, such as the expected number of tasks whose processes can be completed and the task completion probability are investigated under the proposed model.

An infinite-server queue influenced by a semi-Markovian environment

We consider an infinite-server queue, where the arrival and service rates are both governed by a semi-Markov process that is independent of all other aspects of the queue. In particular, we derive a system of equations that are satisfied by various “parts” of the generating function of the steady-state queue-length, while assuming that all arrivals bring an amount of work to the system that is either Erlang or hyperexponentially distributed. These equations are then used to show how to derive all moments of the steady-state queue-length. We then conclude by showing how these results can be slightly extended, and used, along with a transient version of Little’s law, to generate rigorous approximations of the steady-state queue-length in the case that the amount of work brought by a given arrival is of an arbitrary distribution.

PERFORMANCE ANALYSIS BASED ON THE NUMBER OF DEBUGGINGS FOR SOFTWARE SYSTEM WITH PROCESSING TIME LIMIT USING RELIABILITY GROWTH MODEL

We propose the performance evaluation method for the multi-task system with software reliability growth process. The software fault-detection phenomenon in the dynamic environment is described by the Markovian software reliability model with imperfect debugging. We assume that the cumulative number of tasks arriving at the system follows the homogeneous Poisson process. Then we can formulate the distribution of the number of tasks whose processes can be complete within a prespecified processing time limit with the infinite-server queueing model. From the model, several quantities for software performance measurement considering the real-time property can be derived and given as the functions of time and the number of debuggings. Finally, we present several numerical examples of the quantities to analyze the relationship between the software reliability characteristics and the system performance measurement.

Transient Analysis for State-Dependent Queues with Catastrophes

Queueing systems with catastrophes and state-dependent arrival and service rates are considered. For two types of queueing systems namely, queues with discouraged arrivals and infinite server queue, explicit expressions for the transient probabilities of system size are obtained by using continued fractions technique. Some system performance measures and steady-state probabilities are studied. The effect of system parameters on system size probabilities are also illustrated numerically. It is observed that the steady-state probabilities differ when catastrophes are present, while they are identical in the absence of catastrophes.

Analytical Model for Connectivity in Vehicular Ad Hoc Networks

We investigate connectivity in the ad hoc network formed between vehicles that move on a typical highway. We use a common model in vehicular traffic theory in which a fixed point on the highway sees cars passing it that are separated by times with an exponentially distributed duration. We obtain the distribution of the distances between the cars, which allows us to use techniques from queuing theory to study connectivity. We obtain the Laplace transform of the probability distribution of the connectivity distance, explicit expressions for the expected connectivity distance, and the probability distribution and expectation of the number of cars in a platoon. Then, we conduct extensive simulation studies to evaluate the obtained results. The analytical model that we present is able to describe the effects of various system parameters, including road traffic parameters (i.e., speed distribution and traffic flow) and the transmission range of vehicles, on the connectivity. To more precisely study the effect of speed on connectivity, we provide bounds obtained using stochastic ordering techniques. Our approach is based on the work of Miorandi and Altman, which transformed the problem of connectivity distance distribution into that of the distribution of the busy period of an equivalent infinite server queue. We use our analytical results, along with common road traffic statistical data, to understand connectivity in vehicular ad hoc networks.