Nir Perel and Uri Yechiali. An Alternating Queue with Slow Servers and Impatient Customers submission

Abstract. Abstract

We study M/M/c queues (c = 1, 1 < c < infinity, c = infinity) in a two-phase
(fast and slow) Markovian random environment, with impatient
customers when the system is in the slow phase. The system resides in
the fast phase (phase 1) an exponentially distributed random time with
parameter 'eta' and the arrival and service rates are 'lambda' and 'μ', respectively.
The corresponding parameters for the slow phase (phase 0) are 'gamma',
'lambda_0' and 'μ_0' (<= 'μ'). When the system operates in the slow phase,
customers become impatient. That is, each customer, upon arrival, activates
an individual timer, exponentially distributed with parameter 'xi. If
the system does not change its environment from 0 to 1 before the
customer’s timer expires, the customer abandons the queue, never to
return.
For each queue we derive the generating functions of the system’s
steady-state probabilities and calculate (i) the mean total number of
customers in the system, (ii) the mean total sojourn time of an arbitrary
customer, (iii) the mean sojourn time of a served and of an
un-served customer and (iv) the probability that a customer abandons
the queue. Several extreme cases are examined and numerical results
are presented.d decrease in the asymptotic variance rate.  We call this phenomenon, BRAVO (the acronym stands for Balancing Reduces Asymptotic Variance of Outputs) and end with some numerical results related to BRAVO that leave some open questions.