Operations research Assignment Sheet 1

Operations research assignment
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  BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE, PILANI, GOA CAMPUS First Semester 2010 – 2011 AAOC C312 OPERATIONS RESEARCH Assignment Sheet 1(Queueing System) 1.   Consider the queueing system with mean arrival rate ,......2,1,0,  ==  n n  λ λ   and mean service rate ⎪⎩⎪⎨⎧==== ,.....8,7,3 6,5,4,2 3,2,1, nnn n μ μ μ μ   What relationship should exist between λ   and μ   so that steady state probabilities exist? Find the steady state probabilities. 2.   Customers arrive at a one window drive–in bank according to Poisson distribution with mean 10 per hour. Service time per customer is exponential with mean 5 minutes. The space in front of window, including that for the serviced car can accommodate a maximum of three cars. Other cars can wait outside this space. (i) What is the probability that an arriving customer can drive directly to the space in front of the window? (ii) What is the  probability that an arriving customer will have to wait outside the indicated space? (iii) How long is an arriving customer expected to wait before being served? 3.   In a single server queueing system with Poisson input and exponential service time, arrivals tend to get discouraged when more and more customers are present in the system. Assume that ,......2,1,0, 12 =+=  nn n λ  and ,......2,1,3  ==  n n μ   Compute steady-state probability distribution,  L , and W  . 4.   Babies are born in a sparsely populated state at the rate of one birth every 12 minutes. The time between births follows an exponential distribution. Find the following: (i) The average number of births per year. (ii) The  probability that no births will occur in any one day. (iii) The probability of issuing 50 birth certificates in 3 hours given that 40 certificates were issued during the first 3 hours of the 3 hours period. 5.   B&K Groceries operates with three checkout counters. The manager uses the following schedule to determine the number of counters in operation depending on the number of customers in the store:  No. of customers in the store No. of counters in operations 1-3 1 4-6 2 More than 6 3 Customers arrive in the counter area according to a Poisson distribution with a mean rate of 10 customers per hour. The average check out time per customer is exponential with the mean 12 minutes. Determine the steady-state probability P n  of n  customers in the checkout area. 6.   Visitors’ parking at Ozark College is limited to five spaces only. Cars making use of this space arrive according to a Poisson distribution at the rate of 6 cars per hour. Parking time is exponentially distributed with a mean of 30 minutes. Visitors who cannot find an empty space immediately on arrival may temporarily wait inside to lot until a parked car leaves. That temporary space can hold only three cars. Other cars that cannot park or find a temporary waiting space must go elsewhere. Determine the following: (i) The probability P n  of having n  cars in the system. (ii) The effective arrival rate for cars that actually use the lot. (iii) The average number of cars in the lot. (iv)The average time a car wait for a parking space inside the lot. (v) The average number of occupied  parking space. (vi) The average utilization of parking lot. 7.   Automata car wash facility operates with only one bay. Cars arrive according to a Poisson distribution with a mean of 4 cars per hour and may wait in the facility’s parking lot if the bay is busy. Time for washing and cleaning a car is exponential, with a mean of 10 minutes. Cars that cannot park in the lot can wait in the street  bordering the wash facility. Determine the size of the parking lot such that an arriving car will find a parking space at least 90% of the time.  8.   Consider a bank with two tellers. In average 80 customers per hour arrive at the bank and wait in a single line for an idle teller. The average time it takes to serve a customer is 1.2 minutes. Assume that interarrival times and service times is exponential. Determine (i) The expected number of customers present in the bank; (ii) The expected length of time a customer spends in the bank; (iii) The fraction of time that a particular teller is idle. 9.   An automobile emission inspection station has 3 inspection stalls, each with room for only one car. The station can accommodate at most 4 cars waiting at one time. The arrival pattern is Poisson with a mean of one car every minute during the peak periods. The service time is exponential with mean 6 minutes. The chief inspector wishes to know (i) the average number of cars in the system during the peak hours; (ii) the average waiting time in the system; (iii) the expected number of cars per hour that cannot enter in the station because of full capacity. 10.   The Aditya Township Police Department has 5 patrol cars. A patrol car breaks down and requires service once every 30 days. The police department has two repair workers, each of whom takes an average of 3 days to repair a car. Break down times and repair times are exponential. (i) Determine the average number of police cars in good condition. (ii) Find the average break down time for a police car that needs repairs. (iii) Find the fraction of the time a particular repair worker is idle.
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