TY - JOUR
T1 - The age of the arrival process in the G/M/1 and M/G/1 queues
AU - Haviv, Moshe
AU - Kerner, Yoav
PY - 2011/2
Y1 - 2011/2
N2 - This paper shows that in the G/M/1 queueing model, conditioning on a busy server, the age of the inter-arrival time and the number of customers in the queue are independent. The same is the case when the age is replaced by the residual inter-arrival time or by its total value. Explicit expressions for the conditional density functions, as well as some stochastic orders, in all three cases are given. Moreover, we show that this independence property, which we prove by elementary arguments, also leads to an alternative proof for the fact that given a busy server, the number of customers in the queue follows a geometric distribution. We conclude with a derivation for the Laplace Stieltjes Transform (LST) of the age of the inter-arrival time in the M/G/1 queue.
AB - This paper shows that in the G/M/1 queueing model, conditioning on a busy server, the age of the inter-arrival time and the number of customers in the queue are independent. The same is the case when the age is replaced by the residual inter-arrival time or by its total value. Explicit expressions for the conditional density functions, as well as some stochastic orders, in all three cases are given. Moreover, we show that this independence property, which we prove by elementary arguments, also leads to an alternative proof for the fact that given a busy server, the number of customers in the queue follows a geometric distribution. We conclude with a derivation for the Laplace Stieltjes Transform (LST) of the age of the inter-arrival time in the M/G/1 queue.
KW - Age of inter arrival time
KW - G/M/1 queue
KW - M/G/1 queue
UR - http://www.scopus.com/inward/record.url?scp=79751530775&partnerID=8YFLogxK
U2 - 10.1007/s00186-010-0337-y
DO - 10.1007/s00186-010-0337-y
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AN - SCOPUS:79751530775
SN - 1432-2994
VL - 73
SP - 139
EP - 152
JO - Mathematical Methods of Operations Research
JF - Mathematical Methods of Operations Research
IS - 1
ER -