Computation of packet response time via Lagrange series expansion

Packets are processed in bulk at the nodes of a computer communication network. A complete analytical solution of the packet response time in a node processor is derived. The node processor is modelled as an M/M^[K]/1 queue. In this queue, the packets arrive according to a Poisson process, and the service times are exponentially distributed. In each service period the number of packets served is the minimum of K and the number of packets present in the queue. In existing literature it is assumed implicitly that a root of a polynomial equation, lying in the interval (0, 1), will be evaluated numerically. A Lagrange series expansion of a root of this polynomial equation of degree (K+1) is derived. The proposed solution technique can be used for evaluating analytically other Markovian bulk service queues, and also the E_r/M/1 queue