Batch service has a wide application in manufacturing, communication networks, and cloud computing. In batch service queues with limited resources, one critical issue is to properly schedule the service so as to ensure the quality of service. In this paper we consider an M/G[a,b]/1/N batch service queue with bulking threshold a, max service capacity b, and buffer capacity N, where N can be finite or infinite. Through renewal theory, busy period analysis and decomposition techniques, we demonstrate explicitly how the bulking threshold influences the system performance such as the mean waiting time and time-averaged number of loss customers in batch service queues. We then establish a necessary and sufficient condition on the optimal bulking threshold that minimizes the expected waiting time. Enabled by this condition, we propose a simple algorithm which guarantees to find the optimal threshold in polynomial time. The performance of the algorithm is also demonstrated by numerical examples.

This paper analyzes a discrete-time finite buffer renewal input queue with multipleworking vacations where services are performed in batches of maximum size “b”. The service times bothduring a regular service period and vacation period and vacation times are geometricallydistributed. Employing the supplementary variable and imbedded Markov chain techniques, wederive the steady-state queue length distributions at pre-arrival, arbitrary and outsideobserver’s observation epochs. Based on the queue length distributions, some performancemeasures and waiting time distribution in the queue have been discussed. Finally,numerical results showing the effect of model parameters on the key performance measuresare presented.

In this paper, a batch arrival general bulk service queueing system with interruptedvacation (secondary job) is considered. At a service completion epoch, if the server findsat least ‘a’ customers waiting for service say ξ, heserves a batch of min (ξ, b) customers, whereb ≥ a. On the other hand, if the queue length is atthe most ‘a-1’, the server leaves for a secondary job (vacation) ofrandom length. It is assumed that the secondary job is interrupted abruptly and the serverresumes for primary service, if the queue size reaches ‘a’, during thesecondary job period. On completion of the secondary job, the server remains in the system(dormant period) until the queue length reaches ‘a’. For the proposedmodel, the probability generating function of the steady state queue size distribution atan arbitrary time is obtained. Various performance measures are derived. A cost model forthe queueing system is also developed. To optimize the cost, a numerical illustration isprovided.

Stochastic monotonicity properties for various classes of queueing networks have been established in the literature mainly with the use of coupling constructions. Miyazawa and Taylor (1997) introduced a class of batch-arrival, batch-service and assemble-transfer queueing networks which can be thought of as generalized Jackson networks with batch movements. We study conditions for stochastic domination within this class of networks. The proofs are based on a certain characterization of the stochastic order for continuous-time Markov chains, written in terms of their associated intensity matrices.

Miyazawa and Taylor (1997) introduced a class of assemble-transfer batch service queueing networks which do not have tractable stationary distribution. However by assuming a certain additional arrival process at each node when it is empty, they obtain a geometric product-form stationary distribution which is a stochastic upper bound for the stationary distribution of the original network. In this paper we develop a stochastic lower bound for the original network by introducing an additional departure process at each node which tends to remove all the customers present in it. This model in combination with the aforementioned upper bound model gives a better sense for the properties of the original network.

We introduce a batch service discipline, called assemble-transfer batch service, for continuous-time open queueing networks with batch movements. Under this service discipline a requested number of customers is simultaneously served at a node, and transferred to another node as, possibly, a batch of different size, if there are sufficient customers there; the node is emptied otherwise. We assume a Markovian setting for the arrival process, service times and routing, where batch sizes are generally distributed.

Under the assumption that extra batches arrive while nodes are empty, and under a stability condition, it is shown that the stationary distribution of the queue length has a geometric product form over the nodes if and only if certain conditions are satisfied for the extra arrivals. This gives a new class of queueing networks which have tractable stationary distributions, and simultaneously shows that the product form provides a stochastic upper bound for the stationary distribution of the corresponding queueing network without the extra arrivals.

We consider the Mk/M/∞ queue with k heterogeneous customers in a batch where the customer of type i in a batch requires an exponential service time with parameter µi. In steady state, the joint generating function of the number of customers of type i being served in the system is derived explicitly by solving a partial differential equation.

A general framework of continuous-time queueing networks is studied with simultaneous state dependent service completions such as due to concurrent servicing or discrete-time slotting and with state dependent batch routings such as most typically modelling blocking. By using a key notion of group-local-balance, necessary and sufficient conditions are given for the stationary distribution to be of product form. These conditions and a constructive computation of the product form are based upon merely local solutions of the group-local-balance equations which can usually be solved explicitly for concrete networks. Moreover, a decomposition theorem is presented to separate service and routing conditions. General batch service and batch routing examples yielding a product form are hereby concluded. As illustrated by various examples, known results on both discrete- and continuous-time queueing networks are unified and extended.

A direct proof is presented for the fact that the stationary system queue length distribution just after the service completion epochs in the Mx/GI/1/k queue is given by the truncation of a measure on Z+ = {0, 1, ·· ·}. The related truncation formulas are well known for the case of the traffic intensity ρ < 1 and for the virtual waiting time process in M/GI/1 with a limited waiting time (Cohen (1982) and Takács (1974)). By the duality of GI/MY/1/k to Mx/GI/1/k + 1, we get a similar result for the system queue length distribution just before the arrival of a customer in GI/MY/1/k. We apply those results to prove that the loss probabilities of Mx/GI/1/k and GI/MY/1/k are increasing for the convex order of the service time and interarrival time distributions, respectively, if their means are fixed.