A study on bulk arrival non Markovian single service queue with state dependent service

This paper concentrates on single server bulk arrival queue system with batch size dependent service and working vacation. The server provides service in two service modes depending upon the queue length. The server provides single service if the queue length is at least ‘a’. On the other hand the server provides fixed batch service if the queue length is at least ‘k’ (k > a). Batch service is provided with some fixed batch size ‘k’. After completion of service if the queue length is less than ‘a’ then the server leaves for working vacation. During working vacation customers are served with lower service rate than the regular service rate. Service during working vacation also contains two service modes. For the proposed model probability generating function of the queue length at an arbitrary time will be obtained by using supplementary variable technique. Some performance measures will also be presented with suitable numerical illustrations.

In this paper, we analyze finite buffer M / G / 1 queue where service is offered in groups/batches according to ‘general bulk service’ rule by a single server. The service time distribution is considered to be generally distributed and allowed to change dynamically depending on the batch size under service and queue length just before the service initiation of the batch under consideration. Using the embedded Markov chain technique and supplementary variable technique, we obtain the joint distribution of the queue length and batch size at various epoch. At the end, we present several numerical results in the form of self-explanatory table and graphs to bring out some interesting features of the model.

This research article investigates the stationary N-policy with a set-up time of a single server bulk arrival queueing model rendering two-heterogeneous optional repeated service (THORS) with the factor of unreliability. The server studied here is allowed to take a single vacation after each busy period following Bernoulli Schedule (BS). An extensive analysis of the proposed model is carried out using the Supplementary Variable Technique (SVT) under some suitable transformations. The joint distribution of the server’s state and queue length are obtained under both elapsed and remaining times along with their double transforms. Probability generating function (PGF) of the queue size distribution together with the mean system size of the model are determined for any arbitrary time point and service completion epoch, besides the busy period distribution and various pivotal system characteristics. A suitable linear cost structure of the underlying model is developed, and with the help of a difference operator, a locally optimal N-policy at a lower cost is obtained. Finally, some numerical experiments are carried out in support of the theory.

This paper considers an infinite buffer single server batch service queue with single exponential working vacation policy. The inter-arrival times are generally independent and identically distributed random variables and the service times are exponential. The server accesses new arrivals even after service has started on any batch of initial number a. This operation continues till the service time of the ongoing batch is completed or the maximum accessible limit d of the batch being served is attained whichever occurs first. The supplementary variable technique and the recursive method are used to develop the steady-state queue length distributions at pre-arrival and arbitrary epochs. Some performance measures and numerical results are discussed.

For a single server queue with state dependent Markovian input and state dependent general service times we give a stable recursive algorithm to compute the steady-state probabilities. An integral representation of the coefficients in the recursive scheme is given. For service time distributions of phase type the computation of the coefficients simplifies considerably in that only matrix-vector operations are needed and no integrations.

We presented a MX /G1, G2 /1 queueing system with a Bernoulli schedule that includes service disruption, working breakdown, balking, catastrophe and extended server vacations. All customers that arrive are served by a single server, with service time determined by general distribution. As soon as the system fails, the server keeps serving the current client at a reduced rate while repairs are made. It is believed that the vacation schedule is broad. Random disruptions to the server occur, with an exponential distribution in the length of the disruption. We assume that besides a Poisson stream of positive arrivals, there is a Poisson stream of negative arrivals, which we refer to as catastrophes. Using the supplementary variable technique, the Laplace transforms of the time-dependent probability of the system state are derived. We can infer steady-state outcomes from this. The average waiting time and queue size were also obtained. An additional topic of discussion is Adaptive Neuro-Fuzzy Inference system (ANFIS). Furthermore, this work uses Particle Swarm Optimisation (PSO), Artificial Bee Colonies (ABC), and Genetic Algorithms (GA) to swiftly find the system’s optimal cost. Additionally, we examined the convergence of several graph-optimisation strategies.

This paper studies a single server queue with variable batch size general service, Poisson bulk arrival and multiple working vacations where the server works with different rate rather than completely stops the service during the vacation period. In this model, the servers with variable batch size (a, b) in regular service period whereas, during the working vacation period the server starts the service if he finds even one customer in queue with a maximum of 'b' customers with different service time. Service time in working vacation and in regular period follows general distribution. The probability generating function of a queue size at an arbitrary time epochs as well as other completion epochs are derived. Expected queue length in a steady state is also obtained. Numerical illustration is presented.

The objective of this paper is to analyze an $$M^{[X]}/G(a,b)/1$$ queueing model with admission control, multiple vacation, restricted re-service, setup and closedown times. At the moment of service completion, the leaving batch of customers may request for re-service with probability $$\eta $$ and it is not compulsory to accept the request. The server accepts the request with probability $$\zeta $$ and restricts it with probability $$1-\zeta $$ . All the batches of customers are not allowed all the time into the system. On completion of a service or re-service, if the queue length is $$\omega $$ , where $$\omega <a$$ , then the server resumes closedown work. After that, the server leaves for a vacation of random length. When the server returns from a vacation if the queue length is still less than a, he leaves for another vacation and so on until he finds at least a customers in the queue. If the queue length is at least a, then the server commences setup work before starting the service. Using supplementary variable technique, the steady-state probability generating function of the system size at an arbitrary time is obtained. The performance measures and cost model are also derived. Numerical illustrations are presented to visualize the effect of system parameters.

Preface to the Second Edition. Preface to the First Edition. 1. Introduction. 1.1 Motivation. 1.2 Methodological Background. 1.3 Basics of Probability and Statistics. 2. Markov Chains. 2.1 Markov Processes. 2.2 Performance Measures. 2.3 Generation Methods. 3. Steady-State Solutions of Markov Chains. 3.1 Solution for a Birth Death Process. 3.2 Matrix-Geometric Method: Quasi-Birth-Death Process. 3.3 Hessenberg Matrix: Non-Markovian Queues. 3.4 Numerical Solution: Direct Methods. 3.5 Numerical Solution: Iterative Methods. 3.6 Comparison of Numerical Solution Methods. 4. Steady-State Aggregation/Disaggregation Methods. 4.1 Courtois' Approximate Method. 4.2 Takahashi's Iterative Method. 5. Transient Solution of Markov Chains. 5.1 Transient Analysis Using Exact Methods. 5.2 Aggregation of Stiff Markov Chains. 6. Single Station Queueing Systems. 6.1 Notation. 6.2 Markovian Queues. 6.3 Non-Markovian Queues. 6.4 Priority Queues. 6.5 Asymmetric Queues. 6.6 Queues with Batch Service and Batch Arrivals. 6.7 Retrial Queues. 6.8 Special Classes of Point Arrival Processes. 7. Queueing Networks. 7.1 Definitions and Notation. 7.2 Performance Measures. 7.3 Product-Form Queueing Networks. 8. Algorithms for Product-Form Networks. 8.1 The Convolution Algorithm. 8.2 The Mean Value Analysis. 8.3 Flow Equivalent Server Method. 8.4 Summary. 9. Approximation Algorithms for Product-Form Networks. 9.1 Approximations Based on the MVA. 9.2 Summation Method. 9.3 Bottapprox Method. 9.4 Bounds Analysis. 9.5 Summary. 10. Algorithms for Non-Product-Form Networks. 10.1 Nonexponential Distributions. 10.2 Different Service Times at FCFS Nodes. 10.3 Priority Networks. 10.4 Simultaneous Resource Possession. 10.5 Prograrns with Internal Concurrency. 10.6 Parallel Processing. 10.7 Networks with Asymmetric Nodes. 10.8 Networks with Blocking. 10.9 Networks with Batch Service. 11. Discrete-Event Simulation. 11.1 Introduction to Simulation. 11.2 Simulative or Analytic Solution? 11.3 Classification of Simulation Models. 11.4 Classification of Tools in DES. 11.5 The Role of Probability and Statistics in Simulation. 11.6 Applications. 12. Performance Analysis Tools. 12.1 PEPSY. 12.2 SPNP. 12. 3 MOSEL-2. 12.4 SHARPE. 12.5 Characteristics of Some Tools. 13. Applications. 13.1 Case Studies of Queueing Networks. 13.2 Case Studies of Markov Chains. 13.3 Case Studies of Hierarchical Models. Glossary. Bibliography. Index.

In this paper, we used the accepted methodology of applying probabilistic-model checking to dynamic power management. For the first time, we analyze in this context a dynamic-power-management system with the two request priorities. First system we consider has a single two-priority service request queue. Second system to be analyzed has two single service request queues that are dedicated for high- and low-priority requests. We calculate optimal stochastic policies for the both systems. The same delay constraint is applied on the whole queue of the first system and on the high-priority (QH) queue of the second system. The comparison of the power consumptions of the two systems shows that the second system consumes up to 77% less power. We obtain that the power consumptions of both systems converge when the constraints become tighter. We demonstrate that, with the increased length of the QH service request queue of the second system, more power can be saved in applications with varying quality-of-service requirements. Our models and experiments can be easily generalized to <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> request priorities and different queue lengths.

In this paper we discuss the performance analysis of a single server queue with Poisson arrivals and arbitrary service times which are conditioned on the number of customers (units) in the system at the beginning of the service of a customer. The system has a finite capacity K. The model is denoted by M/Gn/K, and has wide applications in computer-communication systems. An algorithm has been developed to compute the steady-state probabilities at departure and arbitrary epochs. Numerical results have been presented for various special cases: two-state service rates; two-state mixed service; and many-state service rates. The algorithm is computationally efficient, accurate and reliable for both high and low values of the model parameters and works for all the service time distributions including the non-phase type. Some comparative studies have been presented in the form of tables and graphs.

In cooperative content distribution (CCD), every Mobile Terminal (MT) cooperate with each other to distribute the content to those terminals which are within their transmission range and are interested in the same content. It has been already proved that CCD is more energy efficient than the traditional approach where each MT download the whole content individually. In this paper, queuing theory has been applied with CCD schemes, different CCD models have been compared with each other and its performance under different queuing models has been analyzed. Simulation result shows that Single Server Multiple Queue (SSMQ) and Multiple Server Multiple Queue (MSMQ) are more efficient in terms of energy consumption as compare to Single Server Single Queue (SSSQ) and Multiple Server Single Queue (MSSQ).

This paper presents a discrete time bulk service queue under the policy (a, c, d) with state dependent service rates. The inter-arrival times are assumed to be independent and geometrically distributed. The customers are served by a single server under the policy (a, c, d) with state dependent service rates. In this model the server begins service only when there are at least c units in the queue and he serves a maximum of d units in a batch. The server continue to serve even when the queue size is less than c but not less than a secondary limit a after a service completion epoch, but with a different service rate. If after a service completion epoch the queue size is less than a, the server becomes idle. The service times are also assumed to be independent and geometrically distributed. The steady state probabilities and some performance measures of the model are computed.

This paper considers a single server retrial queue in which a state-dependent service policy is adopted to control the service rate. Customers arrive in the system according to a Poisson process and the service times and inter-retrial times are all exponentially distributed. If the number of customers in orbit is equal to or less than a certain threshold, the service rate is set in a low value and it also can be switched to a high value once this number exceeds the threshold. The stationary distribution and two performance measures are obtained through the partial generating functions. It is shown that this state-dependent service policy degenerates into a classic retrial queueing system without control policy under some conditions. In order to achieve the social optimal strategies, a new reward-cost function is established and the global numerical solutions, obtained by Canonical Particle Swarm Optimization algorithm, demonstrate that the managers can get more benefits if applying this state-dependent service policy compared with the classic model.

The management of large complex systems is complicated by self-regulation—the phenomenon whereby changes in one component lead to compensating changes elsewhere. One characteristic of such systems (though by no means inevitable) is the potential for chaos—an unstable dynamic that makes predictions impossible. Papadopoulos et al.1, in the JRSM, and Smethurst and Williams2, in Nature, have argued that National Health Service (NHS) waiting lists show self-regulatory behaviour and may be capable of chaotic behaviour. These workers plotted the frequency distribution of the relative changes in queue size on a double logarithmic scale. When the double logarithmic plots appeared linear, the so-called power law was taken as evidence for self-regulation. We believe this technique to be fundamentally flawed. Unregulated time series can show power laws and we have suggested alternative approaches3. Here we apply appropriate tests to data on the size of waiting lists in 10 randomly chosen hospitals over the period 1998-2001 (data taken from [www.doh.gov.ac.uk ]). To each dataset we applied three tests. First, to check for long-term trends (e.g. increases or decreases in waiting lists), we performed linear regressions of queue size on time. Second, we estimated slope of the relation between change in log queue size and log queue size to see if this was negative. This test looks for evidence that large queues tend to get smaller whilst small queues get larger, as in self-regulated systems. Statistical significance tests cannot be performed on the slopes of such regressions, since for random time series the slope of this relation is biased and is expected to be negative. Instead we were interested in whether slopes were steeper than - 1, the implication of this value being that chaos is only possible if the slope is steeper than - 1. Third, we tested for self-regulation using a test (the Pollard test) explicitly designed to detect self-regulation in time series4. The results are shown in Table 1. 6 out of 10 time series showed evidence for long-term trends, all negative, indicating significant declines in queue length. The slopes estimated from the regressions of change in log queue size on queue size were steeper than - 1 only in one case, and this was only slightly steeper (- 1.11). As noted, statistical tests on these slopes are biased. The Pollard test, which corrects for this bias, indicated that only one time series exhibited statistically significant evidence for self-regulation; moreover, in view of the number of tests (10), the marginal significance of this relation (P=0.03) should be viewed with caution. Table 1 Analysis of data on waiting lists from 10 randomly chosen hospitals for evidence of changes consistent with chaos or density dependence We do not regard the analysis we have presented as definitive; for instance, longer time series may show different behaviour. However, claims that the NHS is ‘at the edge of chaos’ are not supported by our analyses. We have shown that, for the data in question, there is no evidence of self-regulatory behaviour. Moreover, waiting lists do show evidence of clear declines, probably resulting from management strategies to reduce queue sizes.