Question

Question 1.
a) Consider a queuing system that consists of n machines working in series (serial system). That is, each job in
the queue has to go through all machines. The lifetime of each machine is exponentially distributed with
rate li (i=1,…n). What is the expected amount of time until the failure of the system?
b) Consider a queuing system with n parallel machines (parallel system). That is, each job in the queue can be
completed by one of the machines in the system. The lifetime of each machine is exponentially distributed
with l. What is the expected amount of time until the system failure (break down)?

Answer 1

To justify this queuing system machines working under overview of important contribution on the area of machine repair model in different framework.

Machines are integral part of all industries, which also cause of certainty to failure.

The operation of any machining system may be stopped due to the failure of the machines and in such a situation, the repairs facility should be available to repair of the failed machines.

Finite source of queuing system where machines fail at irregular Interval and are sent for repairing to repair facility having one or more repairmen. When are repairmen are busy and the system is failed, then production stops. Whenever machine fails, it result in a loss to the organization in terms of production consequently in money.

Time concept can play significant role to regulate the service provided by machines. We consider a machine repair problem wherein M operating machine and R repairmen are available and the repair term are identical exponentially distributed random variables with mean.

The processing time requirements of each job are exponentially distributed with service rate. The single processor service to jobs with different rate depending upon the number of jobs present in the system

If suppose where they are less then n jobs only service position are available for the jobs. When there are greater than jn and less than or equal to (j+1)n jobs then (R+j) (j=1,2....r-1) service position are available for providing service.

It would also be helpful in preventing the failed machines from being discouraged and leaving the system. Various aspects of the servers including vacation and break down have treated to account for the situation of real time machining system wherein server are subject to breakdown.