A single-processor system in which all jobs present simultaneously share the fixed capacity of the processor equally, is considered. Defined is the processing time of a job as the service load that it carries with it on arrival into the system. It is supposed that jobs arrive into the system in a Poisson stream with rate \(\lambda\) and that their processing times are independent and identically distributed with distribution function F and mean 1/\(\mu\). It is assumed also \(\lambda<\mu\) and that the system is in equilibrium.
In this system the response time of a job as its total time in the system is defined. The conditional expected response time \(R_ n(t)\) of a job that requires processing time t and meets n jobs in the system on arrival is investigated. Let \(H(x)=\mu \int^{x}_{0}(1-F(y))dy\) and \(H_ j\), \(j=0,1,..\). be the j-fold convolution of H with \(H_ 0(\cdot)=1\). The main result of the paper is given by the formula \[ (1)\quad R_ n(t)=t/(1-\rho)+(n-\rho /(1-\rho))(1/\rho)\int^{t}_{0}(1-W(x))dx \] where \(\rho =\lambda /\mu\) and \(W(x)=(1- \rho)\sum^{\infty}_{j=0}\rho^ jH_ j(x)\). As a consequence of this result the expected response time R(t), conditional only on t, is obtained: \(R(t)=t/(1-\rho).\)
This result has been obtained in M. Sakata, S. Noguchi and J. Oizimi, Oper. Res. 19, 371-385 (1971; Zbl 0247.60061). The result (1) for exponential processing times has been obtained in E. G. Coffman, R. R. Muntz and H. Trotter, J. Assoc. Comput. Mach. 17, 123-130 (1970; Zbl 0197.153).