Consider a first-in first-out single-server queueing system in which customers arrive according to a Poisson process...

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Consider a first-in first-out single-server queueing system in which customers arrive according to a Poisson process with rate 2 per time unit. Each customer requires exponential service time with mean 1/μ time units. The system experiences occasional failures, and when the system fails all the customers in the system are dropped (including the customer currently under service). The time distribution between failures is exponentially distributed with mean 1/y time units. Assume that all the random variables are independent. a) Show that the system can be modeled using a Markov chain and draw a state-diagram. b) Write the balance equations for the state-diagram. c) Assume λ=u-y-1. Find the stationary distribution of the number of customers in the queue. Hint: Guess a solution of the form 7(i)= (1-a)a', where (i) represents the stationary probability of finding i customers in the system and a is a variable which value must be determined. d) Assume 2-μ-y-1. Find the fraction of customers that successfully complete service. Consider a first-in first-out single-server queueing system in which customers arrive according to a Poisson process with rate 2 per time unit. Each customer requires exponential service time with mean 1/μ time units. The system experiences occasional failures, and when the system fails all the customers in the system are dropped (including the customer currently under service). The time distribution between failures is exponentially distributed with mean 1/y time units. Assume that all the random variables are independent. a) Show that the system can be modeled using a Markov chain and draw a state-diagram. b) Write the balance equations for the state-diagram. c) Assume λ=u-y-1. Find the stationary distribution of the number of customers in the queue. Hint: Guess a solution of the form 7(i)= (1-a)a', where (i) represents the stationary probability of finding i customers in the system and a is a variable which value must be determined. d) Assume 2-μ-y-1. Find the fraction of customers that successfully complete service.