1 Mathematics • Queueing Theory In a take-away food joint with a single service counter, customers are served on a first-come-first-served basis. The arrival rate follows a Poisson distribution and service time follows an exponential distribution. If 2 customers arrive every 10 minutes and 18 customers are served per hour, what is the expected number of customers in the joint at any given time? In a take-away food joint with a single service counter, customers are served on a first-come-first-served basis. The arrival rate follows a Poisson distribution and service time follows an exponential distribution. If 2 customers arrive every 10 minutes and 18 customers are served per hour, what is the expected number of customers in the joint at any given time? A. 2 2 B. 5 5 C. 9 9 D. 1 1 उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: A Correct Answer: A For the M/M/1 queue, lambda = 12 per hour and mu = 18 per hour. The expected number in the system is lambda/(mu - lambda) = 12/6 = 2. For the M/M/1 queue, lambda = 12 per hour and mu = 18 per hour. The expected number in the system is lambda/(mu - lambda) = 12/6 = 2.
2 Mathematics • Queueing Theory In a take-away food joint with a single service counter, customers are served on a first-come-first-served basis. Arrivals follow a Poisson distribution and service times follow an exponential distribution. If 2 customers arrive every 10 minutes and 18 customers are served per hour, what is the probability that an incoming customer waits for more than 30 minutes before being served? In a take-away food joint with a single service counter, customers are served on a first-come-first-served basis. Arrivals follow a Poisson distribution and service times follow an exponential distribution. If 2 customers arrive every 10 minutes and 18 customers are served per hour, what is the probability that an incoming customer waits for more than 30 minutes before being served? A. (2/3)e^-4 (2/3)e^-4 B. (2/3)e^-2.5 (2/3)e^-2.5 C. (2/3)e^-3 (2/3)e^-3 D. (2/3)e^-2 (2/3)e^-2 उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: C Correct Answer: C Here lambda = 12 per hour, mu = 18 per hour and rho = 2/3. For an M/M/1 queue, P(Wq > t) = rho e^(-(mu-lambda)t). At t = 0.5 hour this is (2/3)e^-3. Here lambda = 12 per hour, mu = 18 per hour and rho = 2/3. For an M/M/1 queue, P(Wq > t) = rho e^(-(mu-lambda)t). At t = 0.5 hour this is (2/3)e^-3.
3 Mathematics • Queueing Theory In a newly opened bank, customers arrive at the rate of 2 customers per hour and are served by a single teller counter in about 20 minutes. The bank has a newly implemented policy that there should be no more than a 5% chance that a customer has to wait for more than 30 minutes in the queue before being served. How many additional teller counters should be opened to meet the policy requirements? In a newly opened bank, customers arrive at the rate of 2 customers per hour and are served by a single teller counter in about 20 minutes. The bank has a newly implemented policy that there should be no more than a 5% chance that a customer has to wait for more than 30 minutes in the queue before being served. How many additional teller counters should be opened to meet the policy requirements? A. 0 (the existing counter is sufficient) 0 (the existing counter is sufficient) B. 3 3 C. 1 1 D. 2 2 उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: C Correct Answer: C Applying the multi-server queue waiting-time criterion to the stated arrival rate, service rate and 30-minute threshold shows that two counters in total are required. Therefore one additional counter is needed. Applying the multi-server queue waiting-time criterion to the stated arrival rate, service rate and 30-minute threshold shows that two counters in total are required. Therefore one additional counter is needed.
4 Mathematics • Queueing Theory In a car wash with one cleaning machine, if cars arrive at a rate of a1 per minute and the expected service time for each car is b1 minutes, the utilization factor is: In a car wash with one cleaning machine, if cars arrive at a rate of a1 per minute and the expected service time for each car is b1 minutes, the utilization factor is: A. 1/(a1-b1) 1/(a1-b1) B. a1*b1 a1*b1 C. a1/b1 a1/b1 D. a1/(a1-b1) a1/(a1-b1) उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: B Correct Answer: B Utilization is arrival rate divided by service rate. Since mean service time is b1, the service rate is 1/b1, so utilization is a1b1. Utilization is arrival rate divided by service rate. Since mean service time is b1, the service rate is 1/b1, so utilization is a1b1.