60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX)
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Keywords
- busy period (1)
- impatient customers (1)
- infinite-server (1)
- random environment (1)
- sojourn times (1)
- state-dependent processor sharing (1)
- time transformation (1)
- waiting time dependent service (1)
- workload (1)
Institute
The paper deals with the workload and busy period for the M/GI/1
system under FCFS discipline, where the customers may become
impatient during their waiting for service with generally
distributed maximal waiting times and also during their service
with generally distributed maximal service times depending
on the time waited for service. This general impatience mechanism,
originally introduced by Kovalenko (1961) and considered
by Daley (1965), too, covers the special cases of impatience
on waiting times as well as impatience on sojourn times,
for which Boxma et al. (2010), (2011) gave new results
and outlined special cases recently. Our unified approach
bases on the vector process of workload and busy time.
Explicit representations for the LSTs of workload and
busy period are given in case of phase-type distributed
impatience.
We deal with an infinite-server system where the
service speed is governed by a stationary and ergodic
process with countably many states. Applying a random
time transformation such that the service speed
becomes one, the sojourn time of a class of virtual
requests with given required service time is equal
in distribution to an additive functional defined
via a stationary version of the time-changed process.
Thus bounds for the expectation of functions of additive
functionals yield bounds for the expectation
of functions of virtual sojourn times, in particular
bounds for fractional moments and the distribution
function. Interpreting the $GI(n)/GI(n)/\infty$ system or
equivalently the $GI(n)/GI$ system under state-dependent
processor sharing as an infinite-server system with
random states given by the number $n$ of requests
in the system provides results for sojourn times
of virtual requests. In case of $M(n)/GI(n)/\infty$,
the sojourn times of arriving and added requests are
equal in distribution to sojourn times of virtual
requests in modified systems, which yields many results
for the sojourn times of arriving and added requests.
In case of integer moments, the bounds generalize
earlier results for $M/GI(n)/\infty$. In particular,
the mean sojourn times of arriving and added requests
in $M(n)/GI(n)/\infty$ are proportional to the required
service time, generalizing Cohen's famous result
for $M/GI(n)/\infty$.