A Lindley-type equation arising from a carousel problem

Abstract

In this paper we consider a system with two carousels operated by one picker. The items to be picked are randomly located on the carousels and the pick times follow a phase-type distribution. The picker alternates between the two carousels, picking one item at a time. Important performance characteristics are the waiting time of the picker and the throughput of the two carousels. The waiting time of the picker satisfies an equation very similar to Lindley's equation for the waiting time in the P H/U/1 queue. Although the latter equation has no simple solution, we show that the one for the waiting time of the picker can be solved explicitly. Furthermore, it is well known that the mean waiting time in the P H/U/1 queue depends on to the complete interarrival time distribution, but numerical results show that, for the carousel system, the mean waiting time and throughput are rather insensitive to the pick-time distribution.

A non-increasing Lindley-type equation

“…Other work on this recursion includes the work on a two-carousel system by Park et al [15], where the authors derive the steady-state waiting-time distribution assuming that B n is uniformly distributed on [0, 1] and A n is either exponential or deterministic. Keeping the carousel application in mind, in [19] this result is extended, by assuming that A n follows a phase-type distribution, and in [18], by assuming that B n follows a polynomial distribution. Here we would like to complement these results by letting now B n follow some general distribution while the service times A n are exponentially distributed with parameter µ.…”

Time-Dependent Behaviour of an Alternating Service Queue

“…Specifically, in Park et al [13], the goal is to derive the steady-state waiting time distribution under specific assumptions on the distributions of A n and B n that are relevant to the carousel application considered. These results are extended in [14,15,16,17], where the main focus is on the steady-state distribution of the waiting time. Contrary to the above-mentioned work, this paper focuses on the time-dependent behaviour of the process {W n }.…”

Exact solution to a Lindley-type equation on a bounded support

“…Substituting this solution into (11), we obtain a linear system that determines ξ and η. Thus we can obtain the general solution to the differential equation (11). From this point on, by following the same method, we can formulate a linear system that determines the coefficients d i and π 0 , and obtain the solution to (8).…”

“…Furthermore, if F B has a bounded support, then we cannot readily apply previously obtained results. In [11] only the case where F B is the uniform distribution is covered, while the method described in [9] is not applicable (since distributions on a bounded support are excluded from the class of distributions that are considered there).…”

“…The authors are mainly interested in the steady-state waiting time of the picker. This problem is further investigated in [11], where the authors expand the results in [7] by allowing the pick times to follow a phase-type distribution.…”