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We consider a system with Poisson arrivals and i.i.d. service times. The requests are served according to the state-dependent processor sharing discipline, where each request receives a service capacity which depends on the actual number of requests in the system. The linear systems of PDEs describing the residual and attained sojourn times coincide for this system, which provides time reversibility including sojourn times for this system, and their minimal non negative solution gives the LST of the sojourn time $V(\tau)$ of a request with required service time $\tau$. For the case that the service time distribution is exponential in a neighborhood of zero, we derive a linear system of ODEs, whose minimal non negative solution gives the LST of $V(\tau)$, and which yields linear systems of ODEs for the moments of $V(\tau)$ in the considered neighborhood of zero. Numerical results are presented for the variance of $V(\tau)$. In case of an M/GI/2-PS system, the LST of $V(\tau)$ is given in terms of the solution of a convolution equation in the considered neighborhood of zero. For bounded from below service times, surprisingly simple expressions for the LST and variance of $V(\tau)$ in this neighborhood of zero are derived, which yield in particular the LST and variance of $V(\tau)$ in M/D/2-PS.
We consider a system with Poisson arrivals and general service times, where the requests are served according to the State-Dependent Processor Sharing (SDPS) discipline (Cohen's generalized processor sharing discipline), where each request receives a service capacity which depends on the actual number of requests in the system. For this system, denoted by $M/GI/SDPS$, we derive approximations for the squared coefficients of variation of the conditional sojourn time of a request given its service time and of the unconditional sojourn time by means of two-moment fittings of the service times. The approximations are given in terms of the squared coefficients of variation of the conditional and unconditional sojourn time in related $M/D/SDPS$ and $M/M/SDPS$ systems, respectively. The numerical results presented for $M/GI/m-PS$ systems illustrate that the proposed approximations work well.
We consider a system with Poisson arrivals and i.i.d. service times and where the requests are served according to the state-dependent (Cohen's generalized) processor sharing discipline, where each request in the system receives a service capacity which depends on the actual number of requests in the system. For this system we derive asymptotically tight upper bounds for the moments of the conditional sojourn time of a request with given required service time. The bounds generalize corresponding results, recently given for the single-server processor sharing system by Cheung et al. and for the state-dependent processor sharing system with exponential service times by the authors. Analogous results hold for the waiting times.
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 $M/GI(n)/\infty$.
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 $M/GI(n)/\infty$.
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 consider a system with Poisson arrivals and i.i.d. service times and where the requests are served according to the state-dependent (Cohen's generalized) processor sharing discipline, where each request in the system receives a service capacity which depends on the actual number of requests in the system. For this system we derive asymptotically tight upper bounds for the moments of the conditional sojourn time of a request with given required service time. The bounds generalize corresponding results, recently given for the single-server processor sharing system by Cheung et al. and for the state-dependent processor sharing system with exponential service times by the authors. Analogous results hold for the waiting times.