Informatics and Applications

2015, Volume 9, Issue 3, pp 25-31

JOINT STATIONARY DISTRIBUTION OF m QUEUES IN THE N-SERVER QUEUEING SYSTEM WITH REORDERING

  • A. V. Pechinkin
  • R. V. Razumchik

Abstract

The paper considers a continuous-time N-server queueing system with a buffer of infinite capacity and customer reordering. The Poisson flow of customers arrives at the system. Service times of customers at each server are exponentially distributed with the same parameter. Each customer obtains a sequential number upon arrival. The order of customers upon arrival should be preserved upon departure from the system. Customers which violated the order form different queues in the reordering buffer which has infinite capacity. If there are n, n = 1, N, customers in servers, then the latest customer to occupy a server is called the 1st level customer, the last but one - the 2nd level customer, . . . , the first - the nth level customer. Customers in the reordering buffer that arrived between the 1st level and the 2nd level customers, form the queue number one. Customers, which arrived between the 2nd level and the 3rd level customers, form the queue number two, etc. Customers, which arrived between the Nth level and the (N - 1)th level customers, form the queue number (N - l) in the reordering buffer. Mathematical relations in terms of Z-transform, which make it possible to calculate the joint stationary distribution of the number of customers in the buffer, servers, and in the 1st, 2nd, . . . , mth queues (m = 1, N - 1) in the reordering buffer, are obtained.

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