The results of this section have been published in the paper of Csige and Tomkó [ 16 ]. The reason of its introduction is to show the importance of the service discipline.

Let us consider heterogeneous machines with exponentially distributed operating and repair time with parameter and respectively for the th machine . The failures are repaired by a single repairman according to Processor Sharing, FIFO, and Preemptive Priory disciplines. All involved random variables are supposed to be independent of each other.

Let ,denote the number of failed machines at time . Due to the heterogeneity of the machines this information is not enough to describe the behavior of the system because we have to know which machine is under service. Thus let us introduce an -dimensional vector with components indicating the indexes of the failed machines. Hence for using FIFO discipline machine with index is under service. Under Processor Sharing discipline when all machines are serviced by a proportional service rate, that is if then the proportion is the order of indexes is not important, but a logical treatment we order them as . In the case of Preemptive Priority assuming that the smaller index means higher priority we use the same ordering as before mentioning that in this case the machine with the first index is under service since he has the highest priority among the failed machines.

Due to the exponential distributions the process

is a continuous-time Markov where the ordering of depends ot the service discipline.

Since is a finite state Markov chain thus if the parameters , , are all positive then it is ergodic and hence the steady-state distribution exists. Of course this heavily depends on the service discipline.

12.3.1. 12.3.1. The Queue

Let the distribution of the Markov chain be denoted by

.

It is not difficult to see that for this distribition we have

where is the ordering of the indexes and

The steady-state distribution which is denoted by

,

is the solution of the following set of equations

with normalizing condition

where the summation is mean by all possible combinations of the indexes.

The surprising fact is it can be obtained as

where can be calculated from the normalizing condition.

For the FIFO and Preemptive Priority disciplines the balance equations and the solution is rather complicated and they are omitted. The interested reader is referred to the cited paper. However for all cases the performance measures can be computed the same way.

12.3.2. 12.3.2. Performance Measures

1. Utilization of the server

2. Utilization of the machines

Let denote the utilization of machine . Then

where denotes the mean response time for machine , that is the mean time while it is broken, and

is the probability that the ith achine is failed. Thus

and in FIFO case for the main waiting time we have

Furthermore it is easy to see that the mean number of failed machines can be obtained as

In addition

which is the Little's formula for heterogeneous customers. In particular, for homogeneous case we

which was proved earlier.

Various generalized versions of the machine interference problem with heterogeneous machines can be found in Pósafalvi and Sztrik [ 62 ], [ 63 ].

Let us see some sample numerical results for the illustration of the influence of the service disciplines on the main performance measures

Figure 12.3. Numerical results