ON THE JOINT DISTRIBUTION OF THE DEPARTURE INTERVALS IN AN M/G/1/N QUEUE

Abstract

This paper discusses a stationary departure process from the M /G/l/ N queue. Using a Markoy renewal process, we examine the joint density function fk of the k-successive departure intervals. In Section 2, we discuss the covariance of departure intervals. The departure intervals are statistically independent in case of N = 0 or N = 1, but not in case of N = 2 or N = 3. In Section 3, !k in the M/M /1/ N is shown to be a symmetric function of arrival and service rates, and we find that cov( d1 , dk ) is not dependent on lag k, for k ~ N + 1. Further, we prove that the covariance of departure intervals in the dual (reversed) system is equal to one in the original system, for any lag k. 1. Introduction In this paper, we discuss the departure process of a queueing system. In order to examine the covariance of departure process, we consider the joint density function of the k-successive departure intervals in the M/G/l/N queue. Many papers have been published on this subject. Burke (1) and Finch (6) have proved that the departure process in the M/M/l queue is again a Poisson process. Jenkins (9) has discussed the covariance of departure process in the M/EA/l queue. For the M/G/l/N queue, Daley (2) and Daley & Shanbhag (3) have analyzed the departure process. Disney et al. (5), Magalhaes & Disney (11), and Simon & Disney (15) have studied the joint distribution of departure intervals by using a Markov process. Moreover, King (10) has shown that: (1) cov(d1 , dk) = 0 for k ~ 2, in the M/G/I/O and M/ D/I/I queue; (2) cov(d1 , dk ) = 0 for k ~ 3, in the M/G/l/l queue. For the M/M/l queue, Hubbard et al. (8) have noted that a probability P(j, t) is a symmetrical expression with regard to an arrival rate A and a service rate p, where P(j, t) =Pr{ exactly j customers depart from the system during a time interval (O,t)}. Saito (14) has analyzed the departure process in an M/G/s/O queue. Makino (12) has discussed a loss probability for the M/M /1/ N ~ / M/I / 1 tandem queue. Furthermore, Daley (4) and Reynolds (1:1) have surveyed the departure process. Using a Markov process, we examine the joint density function fk(Xl, X2, ... , Xk) ofthe k­ successive departure intervals in the M/G/I/ N queue. In Section 2, we discuss the covariance of departure intervals of lag k. The departure intervals are statistically independent in case of N = 0 or N = 1, but not in case of N = 2 or N = 3. Especially, in the M/G/1/2 queue, we find that the covariance of departure intervals of lag k are represented as a geometric progression: cov(d1 , dk) = ,8~-3cov(dl, d3), for k ~ 3, where ,81 =Pr{exactly one customer arrives at the system during a service time}.