DISCRETE TIME POISSON QUEUES OPERATED BY TWO HETEROGENEOUS SERVERS UNDER 'FIRST COME FIRST SERVED' QUEUE DISCIPLINE

摘要

The proposed discrete-time queueing systems are Geo(λ)/Geo(μ_1)+Geo(μ_2)/2 and Geo(λ)/Geo(μ_1),Geo(μ_2)/2 that have an infinite number of waiting positions with one faster server i.e. server-1 and a slow server i.e. server-2. If the slow server is free, then according to the classical First Come First Served (FCFS) discipline, an incoming customer is assigned to the slow server. Now since this customer is getting the slowest possible service, customers arriving subsequently might clear out of the system earlier by getting service from the faster server. This is clearly a violation of the FCFS principle. Such a violation is greater for greater heterogeneity of the service capacities of the two servers. For such a situation, this paper proposes how a customer might find it preferable to wait for service at the faster server than to go into the slow server without violating of the FCFS principle through queue discipline-Ⅰ and queue discipline-Ⅱ. Further time axis is divided into fixed length intervals or slots. Customers arrive during the consecutive slots, but they can only start service at the beginning of slots. The numbers of customers that arrive in successive slots are independent, identically distributed (i.i.d.) random variables subject to a condition that only one customer can arrive in a slot with probability λ (0 ＜ λ ＜ 1) and that no customer arrive in a slot with probability 1 - λ. Customer service times are integer multiples of the slot length, which implies that customers leave the system at slot boundaries. All customers are served either by server-Ⅰ according to geometric service time distribution with mean rate μ_1 or by server-2 with geometric service time distribution where mean rate is μ_2 ＜ μ_1 The steady state analysis is then discussed and numerical values to the steady state expected number of customers E(N)_(Geo/Geo+Geo/2), and E(N)_(Geo/Geo,Geo/2) have been computed. To check if the proposed queues operate under FCFS rule and thus satisfy the Little's formula λW = E(N), the actual expected waiting times W_(Geo/Geo+Geo/2) and W_(Geo/Geo,Geo/2) of customers in the system are then calculated numerically. A simple comparison study over these numerical measures proves that there is an insignificant difference between E(N)_(Geo/Geo+Geo/2), and E(N)_(Geo/Geo,Geo/2) values and between the values W_(Geo/Geo+Geo/2) and W_(Geo/Geo,Geo/2) due to the fact that the proposed two alternative queue disciplines here minimize violations of the FCFS discipline in the long run. Finally Geo(λ)/Geo(μ_1)+Geo(μ_2)/2 queueing model is applied to model a single computing node as a server to obtain the power consumption and the associated expected cost for a specific set of input values.