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We consider a single server system consisting of $n$ queues with different types of customers and $k$ permanent customers. The permanent customers and those at the head of the queues are served in processor-sharing by the service facility (head-of-the-line processor-sharing). By means of Loynes' monotonicity method a stationary work load process is constructed and using sample path analysis general stability conditions are derived. They allow to decide which queues are stable and moreover to compute the fraction of processor capacity devoted to the permanent customers. In case of a stable system the constructed stationary state process is the only one and for any initial state the system converges pathwise to the steady state.
For the general G/G/1 processor sharing (PS) system a sample path result for the sojourn times in a busy period is proved, which yields a relation between the sojourn times under PS and FCFS discipline. In particular, the result provides a formula for the mean sojourn time in G/D/1-PS in terms of the mean sojourn time in the corresponding G/D/1-FCFS, generalizing known results for GI/M/1 and M/GI/1. Extensions of the formula provide the basis for a two-moment approximation of the mean sojourn time in G/GI/1-PS in terms of a related G/D/1-FCFS.
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 $M/GI(n)/\infty$.
In this paper a unified approach to central and decentral approximations of the distribution function $F(x,n)$ of the sum of $n$ iid random variables with compact support in $I\!\!R_+$ is given. This approach yields direct Edgeworth expansion (especially the Central limit theorem) and indirect Edgeworth expansion (Theorem of Bahadur-Rao, large deviation results) within a unified framework. An approximative inversion of the LST of $F(x,n)$ (approximation of the complex inversion integral over a line by an integral over a proper bounded arc with a proper integrand) allows to get these approximations and moreover explicit error bounds.
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.