A queueing system with vacations afterNservices

Abstract

This paper is devoted to the study of an M/G/1 queue with a particular vacation discipline. The server is due to take a vacation as soon as it has served exactly N customers since the end of the previous vacation. N may be either a constant or a random variable. If the system becomes empty before the server has served N customers, then it stays idle until the next customer arrival. Such a vacation discipline arises, e.g., in production systems and in order picking in warehouses.We determine the joint transform of the length of a visit period and the number of customers in the system at the end of that period. We also derive the generating function of the number of customers at a random instant, and the Laplace-Stieltjes transform of the delay of a customer.

Stochastic bounds for order flow times in parts-to-picker warehouses with remotely located order-picking workstations

“…Figures 7-9 also illustrate that the lower and upper bounds virtually coincide when λ is large enough and that the values of the bounds become insensitive to the exact distribution of the picking and completion times. This can be understood by examining Formulas (25) and 27 Remark 6. Note that the analysis of the order flow time is partially based on a relation with the cyclic queueing network system depicted in Figure 5.…”

“…Remark 3. The policy of serving exactly N customers during a visit has recently also been studied by Boxma et al [25]. The model in this paper differs in two aspects.…”

“…The model in this paper differs in two aspects. First, Boxma et al [25] assume that the length of the intervisit period is independent of the length of the preceding visit period. Secondly, the on-off feature of the arrival process is not incorporated by Boxma et al The papers also differ in the obtained performance measures; whereas Boxma et al deduce the PGF of the system occupancy at various time epochs and the LST of the delay of a customer, this paper is devoted to the delay of a random order, i.e., the order flow time.…”

A Queueing System with Vacations after a Random Amount of Work

“…This paper is in some respects a companion paper of [1], where the server takes a vacation if it has served a random number N of customers, staying idle in the queue when the system becomes empty before N customers have been served. In [1], the following expressions were established: the joint transform of the length of a visit period and the number of customers in the system at the end of that period, the generating function of the number of customers at a random instant and the Laplace-Stieltjes transform of the delay of a customer. The key idea was to exploit a link with a result from Cohen [5] about the transient behavior of the queue length at customer departure epochs in an ordinary M/G/1 queue and to apply contour integration and the Fuhrmann-Cooper decomposition.…”