Q1 A high-precision laser sensor in a robot arm has a lifetime that is exponentially- distributed with an average life of 10,000 hours.

1. What is failure rate of thesensor?
2. What is the variance of the sensor'slifetime?
3. Compute the probability that the sensor lasts more than 15,000hours?
4. Compute the probability the sensor lasts between 5,000 and20,000?
5. If you know that one of the sensors has lasted 7,500 hours without failure, what is the probability that it will last more than the next 4,500hours?

Q2.CallsarrivetoacustomercarecenteraccoridngtoaPossionproccesswitharrival rate of 3 calls per hour. The service time (handilng and answering the call of the customer) of the customer care employee is exponetitally-distibuted with mean 0.20hour.

Case.1: If there is only one customer care employee to answer the calls:

1. Find the notation of the queuing system corresponding tocase.1?
2. Draw the birth-and-death process diagram for this queueingsystem?
3. Find the utilization factor of this servicefacility?
4. Find the steady-state probabilities? Write theexpressions.
5. Find the steady-state expected number of customers in thesystem?
6. Find the steady-state expected number of customers waiting in thequeue?
7. Find the steady-state expected time a customer spends in thesystem?
8. Find the steady-state expected time a customer waits in thequeue?

Case.2: If there are two identical customer care employees to answer the calls:

1. Find the notation of the queuing system corresponding tocase.2?
2. Draw the birth-and-death process diagram for this queueingsystem?
3. Find the utilization factor of this servicefacility?
4. Find the steady-state probabilities? Write theexpressions.
5. Find the steady-state expected number of customers in thesystem?
6. Find the steady-state expected number of customers waiting in thequeue?
7. Find the steady-state expected time a customer spends in thesystem?
8. Find the steady-state expected time a customer waits in thequeue?
   1. Supposethatsubscriberswithalocalinternetserviceproviderpaytheirmonthly billsinasalespointinthecitycenterofRamallah.Assumethattherearetwoidenticalserversin thesalespoint,eachhavinganexponentialservicetimewithrate=1/2perhour.Also,suppose the arrival rates per hour are as follows: 0= 1, 1= 1, 2= 1/2, 3= 1/3 and k= 0, k = 4, 5, 6,......

1. Draw the birth-and-death process diagram for this queueingsystem?
2. Find the steady-state probabilities of this system? (Hint: use the equation inrate

equals out ratebalance equations to find the probabilities).

1. What is the average arrival rate in steadystate?
2. What is the expected number of customers (subscribers) in the system? Hint: again use the basic definition of expected number of customers and the steady state probabilities you found in part(2.)).