Time dependent analysis of a queueing model by formulating a boundary value problem.

*(English)*Zbl 0544.60094
Modelling and performance evaluation methodology, Proc. int. Semin., Paris 1983, Lect. Notes Control Inf. Sci. 60, 504-517 (1984).

Summary: [For the entire collection see Zbl 0539.00025.]

The analysis of queueing models which can be characterized as a random walk in the first quadrant of the plane often leads to the problem of solving a functional equation for a bivariate generating function. Recently, a method has been developed by which a rather general class of such functional equations related to stationary distributions can be solved with the aid of the theory of boundary value problems, see e.g., G. Fayolle, P. J. B. King and I. Mitrani, Adv. Appl. Probab. 14, 295-308 (1982; Zbl 0495.60087).

In the present study we shall show that the same method can be applied to the analysis of the time dependent behaviour of this class of queueing models. For this discussion a relatively simple model with two types of customers, Poissonian arrival streams, paired services and a general service time distribution will be considered. The generating function of the joint queue length distribution at the nth departure instant will be determined. This function forms the starting point for the analysis of the asymptotic behaviour of the process as \(n\to \infty\).

The analysis of queueing models which can be characterized as a random walk in the first quadrant of the plane often leads to the problem of solving a functional equation for a bivariate generating function. Recently, a method has been developed by which a rather general class of such functional equations related to stationary distributions can be solved with the aid of the theory of boundary value problems, see e.g., G. Fayolle, P. J. B. King and I. Mitrani, Adv. Appl. Probab. 14, 295-308 (1982; Zbl 0495.60087).

In the present study we shall show that the same method can be applied to the analysis of the time dependent behaviour of this class of queueing models. For this discussion a relatively simple model with two types of customers, Poissonian arrival streams, paired services and a general service time distribution will be considered. The generating function of the joint queue length distribution at the nth departure instant will be determined. This function forms the starting point for the analysis of the asymptotic behaviour of the process as \(n\to \infty\).

##### MSC:

60K25 | Queueing theory (aspects of probability theory) |

90B22 | Queues and service in operations research |