Complete characterisation of the customer delay in a queueing system with batch arrivals and batch service

Whereas the buffer content of batch-service queueing systems has been studied extensively, the customer delay has only occasionally been studied. The few papers concerning the customer delay share the common feature that only the moments are calculated explicitly. In addition, none of these surveys consider models including the combination of batch arrivals and a server operating under the full-batch service policy (the server waits to initiate service until he can serve at full capacity). In this paper, we aim for a complete characterisation-i.e., moments and tail probabilities - of the customer delay in a discrete-time queueing system with batch arrivals and a batch server adopting the full-batch service policy. In addition, we demonstrate that the distribution of the number of customer arrivals in an arbitrary slot has a significant impact on the moments and the tail probabilities of the customer delay

On single server batch arrival queueing system with balking, three types of heterogeneous service and Bernoulli schedule server vacation

This paper investigates a batch arrival queueing system in which customers arrives at the system in a Poisson stream following a compound Poisson process and the system has a single server providing three types of general heterogeneous services. At the beginning of each service, a customer is allowed to choose any one of the three services and as soon as a service of any type gets completed, the server may take a vacation or may continue staying in the system. The vacation time is assumed to follow a general (arbitrary) distribution and the server vacation is based on Bernoulli schedule under a single vacation policy. During the server vacation period, impatient customers are assumed to balk. This paper described the model as a bivariate Markov chain and employed the supplementary variable technique to find closed-form solutions of the steady state probability generating function of number of customers, the steady state probabilities of various states of the system, the average queue size, the average system size, and the average waiting time in the queue as well as the average waiting time in the system. Further, some interesting special cases of the model are also derived. Keywords Batch Arrivals. Queueing System. Balking. Heterogeneous types of Service. Bernoulli schedule server vacation. Bivariate Markov Processes. MSC2020-Mathematics Subject Classification 34B07, 60G05, 62E1

ANALYSIS OF BULK ARRIVALS IN QUEUEING MODELS

       Present paper surveys the literature on bulk queueing models. The concept of bulk arrivals and bulk services has gained a tremendous significance in present situations. Due to congestion problem everywhere (banks, metro stations, bus stops, railway reservation, traffic … etc.) researchers have to focus their attention to develop models and mechanism to deal with the same. A number of models have been developed in the area of queueing theory incorporating bulk queueing models. These bulk queueing models can be applied to resolve the congestion problems. Through this survey, an attempt has been made to review the work done on bulk queues, modeling various phenomenons. The goal is to provide sufficient information to analysts, managers and industry people who are interested in using queueing theory to model congestion problems and want to locate the details of relevant models

Density profiles of the exclusive queueing process

The exclusive queueing process (EQP) incorporates the exclusion principle into classic queueing models. It can be interpreted as an exclusion process of variable system length. Here we extend previous studies of its phase diagram by identifying subphases which can be distinguished by the number of plateaus in the density profiles. Furthermore the influence of different update procedures (parallel, backward-ordered, continuous time) is determined

Critical behavior of the exclusive queueing process

The exclusive queueing process (EQP) is a generalization of the classical M/M/1 queue. It is equivalent to a totally asymmetric exclusion process (TASEP) of varying length. Here we consider two discrete-time versions of the EQP with parallel and backward-sequential update rules. The phase diagram (with respect to the arrival probability \alpha\ and the service probability \beta) is divided into two phases corresponding to divergence and convergence of the system length. We investigate the behavior on the critical line separating these phases. For both update rules, we find diffusive behavior for small output probability (\beta\beta_c it becomes sub-diffusive and nonuniversal: the exponents characterizing the divergence of the system length and the number of customers are found to depend on the update rule. For the backward-update case, they also depend on the hopping parameter p, and remain finite when p is large, indicating a first order transition.Comment: v2: published versio

Delay analysis of two batch-service queueing models with batch arrivals: Geo(X)/Geo(c)/1

In this paper, we compute the probability generating functions (PGF's) of the customer delay for two batch-service queueing models with batch arrivals. In the first model, the available server starts a new service whenever the system is not empty (without waiting to fill the capacity), while the server waits until he can serve at full capacity in the second model. Moments can then be obtained from these PGF's, through which we study and compare both systems. We pay special attention to the influence of the distribution of the arrival batch sizes. The main observation is that the difference between the two policies depends highly on this distribution. Another conclusion is that the results are considerably different as compared to Bernoulli (single) arrivals, which are frequently considered in the literature. This demonstrates the necessity of modeling the arrivals as batches

Performance analysis of load based M/M/3 transient queueing system with finite capacity

This paper deals with the investigation of M/M/3 queueing model under the provision of service rates of jobs depend upon the load of the jobs arrived in the system. The customers (jobs) arrive in the system in Poisson fashion and they are served in the chronological order of their arrival. Differential equations of transient probability distribution functions by using a transition diagram have been set up. Laplace Transform, probability generating function, and Rouchey’s Theorem have been applied to get the probability of n customer in time t. Various performance indices such as the mean number of customers in the system, expected number of customers in a queue, probability that one has not to wait, expected mean time spent in a system, expected mean time spent in a queue, probability that the queue size being greater than or equal to N have been obtained analytically. Finally, the analytical results are validated graphically with the help of computing software.Publisher's Versio