Activity $1$ In your sheet $1,$ rename it as Herr Barber $$ Compute for the mean arrival rate and inter$-$arrival time

The Dupit Corp. Problem $$ The Dupit Corporation is a longtime leader in the office photocopier marketplace. $$ Dupit$$s service division is responsible for providing support to the customers by promptly repairing the machines when needed. This is done by the company$$s service technical representatives, or tech reps. $$ Current policy: Each tech rep$$s territory is assigned enough machines so that the tech rep will be active repairing machines $($or traveling to the site$)75\%$ of the time. $$ A repair call averages $2$ hours, so this corresponds to $3$ repair calls per day. $$ Machines average $50$ workdays between repairs, so assign $150$ machines per rep

Alternative Approaches to the Problem $$ Approach Suggested by John phixitt: Modify the current policy by decreasing the percentage of time that tech reps are expected to be repairing machines. $$ Approach Suggested by the Vice President for Engineering: Provide new equipment to tech reps that would reduce the time required for repairs. $$ Approach Suggested by the Chief Financial Officer: Replace the current one$$person tech rep territories by larger territories served by multiple tech reps. $$ Approach Suggested by the Vice President for Marketing: Give owners of the new printer$$copier priority for receiving repairs over the company$$s other customers. $21$ The Queueing System for Each Tech Rep $$ The customers: The machines needing repair. $$ Customer arrivals: The calls to the tech rep requesting repairs. $$ The queue: The machines waiting for repair to begin at their sites. $$ The server: The tech rep. $$ Service time: The total time the tech rep is tied up with a machine, either traveling to the machine site or repairing the machine. $($Thus$,$ a machine is viewed as leaving the queue and entering service when the tech rep begins the trip to the machine site.$)$

Notation for Single$$Server Queueing Models $=$ Mean arrival rate for customers. $=$ Expected number of arrivals per unit time. $1=$ expected interarrival time. $=$ Mean service rate $($for a continuously busy server$).=$ Expected number of service completions per unit time. $1=$ expected service time. $=$ the utilization factor. $=$ the average fraction of time that a server is busy serving customers. $=$

The M$/$M$/1$ Model $1$ Assumptions $1.$ Interarrival times have an exponential distribution with a mean of $1.2.$ Service times have an exponential distribution with a mean of $1.3.$ The queueing system has one server. $$ The expected number of customers in the system is L $==(1)()$ The expected waiting time in the system is W L $==(11)()$ The expected waiting time in the queue is W W q $==1()$ The expected number of customers in the queue is $()()221$ L W q q $===$

The M$/$M$/1$ Model $2$ The probability of having exactly n customers in the system is $(1)$ n P n $=$ Thus, $0$ P $=1$ P$1=(1)()22$ P $=1$ : : $$ The probability that the waiting time in the system exceeds t is $()(1)$ for $0$ t P W t e t $=$ The probability that the waiting time in the queue exceeds t is $()(1)$ for $0$ t P W t e t q $$