Optimal Admission Control for Tandem Queues With Loss

We consider a two-station tandem queue loss model where customers arrive to station 1 according to a Poisson process. A gatekeeper who has complete knowledge of the number of customers at both stations decides to accept or reject each arrival. A cost
c
₁
is incurred if a customer is rejected, while if an admitted customer finds that station 2 is full at the time of his service completion at station 1, he leaves the system and a cost
c
₂
is incurred. Assuming exponential service times at both stations, an arbitrary but finite buffer size at station 1 and a buffer size of one at station 2, we show that the optimal admission control policy for minimizing the long-run average cost per unit time has a simple structure. Depending on the value of
c
₂
compared to a threshold value
c
_*
, it is optimal to admit a customer at the time of his arrival either only if the system is empty or as long as there is space at station 1. We also provide the closed-form expression of
c
_*
, which depends on the service rates at both stations, the arrival rate and
c
₁
.