Asymptotic Analysis for a Closed Processor-Sharing System with Switching Times: Normal Usage

Abstract

In a closed processor-sharing system with CPU-switching times, the probability distribution of the waiting time $W,P_\varepsilon ( t ) = \mathbb{P}\{ {W > t|\varepsilon \geqq 0} \}$ where $\varepsilon $ is the switching time is studied. The waiting time is the time needed for a particular job, called a tagged job, to be served by the CPU. The switching time a is the time spent by the CPU to transfer from one job to the next. In this paper, $\varepsilon $ is used as a perturbative parameter. The system consists of a bank of $N + 1$ terminals, with N large, in series with a CPU, which feeds back to the terminals. Let $p/q$ be the ratio of the mean required service time to the mean think time. The state of the system is characterized by the traffic intensity $\rho = Np/q = O( 1 )$. Let $\pi _i ( \varepsilon )$ be the stationary probabilities, i.e., the probabilities of having i jobs $( i = 0,1, \cdots ,N )$ requiring service by the CPU, when the system is in the steady state. We write $\pi _i ( \varepsilon )$ as a polynomial in $\varepsilon $. In order to compute the equilibrium distribution $P_\varepsilon ( t )$, the first two moments $E( W )$and $E( {W^2 } )$ and the variance $\sigma ^2 ( W )$ for the waiting time, two ordinary differential equations of the third order are written down and solved for normal usage $( \rho < 1 )$ by means of asymptotic expansions in $1/N$ and $\varepsilon $. The first perturbative terms in $1/N$ and $\varepsilon $ are computed for $P_\varepsilon ( t ),E( W ),E( {W^2 } )$, and $\sigma ^2 ( W )$. It is noted that the introduction of a nonzero switching time (that is, $\varepsilon > 0$) implies an increase in the expectation value $E( W )$ and in the variance $\sigma ^2 ( W )$ of the waiting time W. Finally, numerical results are used to illustrate experimentally the range of validity of the approximate expressions that have been obtained.