4,386 research outputs found
Stationary remaining service time conditional on queue length
In Mandelbaum and Yechiali (1979) a simple formula is derived for the expected station-ary remaining service time in a FIFO M/G/1 queue, conditional on the number of customers in the system being equal to j, j ≥ 1. Fakinos (1982) derived a similar formula using an alternative method. Here we give a short proof of the formula using rate conservation law (RCL), and generalize to handle higher moments which better illustrates the advantages of using RCL
Stationary analysis of a single queue with remaining service time dependent arrivals
We study a generalization of the system (denoted by ) with
independent and identically distributed (iid) service times and with an arrival
process whose arrival rate depends on the remaining service
time of the current customer being served. We derive a natural stability
condition and provide a stationary analysis under it both at service completion
times (of the queue length process) and in continuous time (of the queue length
and the residual service time). In particular, we show that the stationary
measure of queue length at service completion times is equal to that of a
corresponding system. For we show that the continuous time
stationary measure of the system is linked to the system via a
time change. As opposed to the queue, the stationary measure of queue
length of the system at service completions differs from its marginal
distribution under the continuous time stationary measure. Thus, in general,
arrivals of the system do not see time averages. We derive formulas
for the average queue length, probability of an empty system and average
waiting time under the continuous time stationary measure. We provide examples
showing the effect of changing the reshaping function on the average waiting
time.Comment: 31 pages, 3 Figure
Stationary distributions of the multi-type ASEP
We give a recursive construction of the stationary distribution of multi-type
asymmetric simple exclusion processes on a finite ring or on the infinite line
. The construction can be interpreted in terms of "multi-line diagrams" or
systems of queues in tandem. Let be the asymmetry parameter of the system.
The queueing construction generalises the one previously known for the totally
asymmetric () case, by introducing queues in which each potential service
is unused with probability when the queue-length is . The analysis is
based on the matrix product representation of Prolhac, Evans and Mallick.
Consequences of the construction include: a simple method for sampling exactly
from the stationary distribution for the system on a ring; results on common
denominators of the stationary probabilities, expressed as rational functions
of with non-negative integer coefficients; and probabilistic descriptions
of "convoy formation" phenomena in large systems.Comment: 54 pages, 4 figure
The Mx/G/1 queue with queue length dependent service times
We deal with the MX/G/1 queue where service times depend on the queue length at the service initiation. By using Markov renewal theory, we derive the queue length distribution at departure epochs. We also obtain the transient queue length distribution at time t and its limiting distribution and the virtual waiting time distribution. The numerical results for transient mean queue length and queue length distributions are given.Bong Dae Choi, Yeong Cheol Kim, Yang Woo Shin, and Charles E. M. Pearc
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