Consider a modified version of an M/M/1 queue (arrival rates are "lambda" , service rates are "mu" ) in which each customer waiting in queue (but not in service) may get impatient and leave will, independently of all others, leave after an exponentially distributed amount of time with parameter "gamma" .
a) Model the queue as a B&D process. Hint: When there exists "n" customers in the system, there is one customer being served and there are -1 customers that may get impatient in the system. The first of all these n events makes the number of customers n-1.
b)Let "gamma" equal to "mu" ,
i.Determine the stationary probability distribution of the number of customers in the system.
ii. Determine the mean number of customers in the system.
iii. Determine the mean time in system for the customers in the system.
iv. Determine the mean number of customers waiting in the queue.
v. Determine the mean waiting time of the customers in the queue.
vi. Find the probability that an arriving customer finds k customers in the system.