3232
ISyE 3232
J. Reed
Stochastic Manufacturing and Service Systems
Fall 2005
Homework 7
November 22, 2005
Due at the start of class on Thursday, December 1st.
1. Suppose there are two tellers taking customers in a bank. Service times at a teller are independent, exponentially distributed random variables, but the first teller has a mean service time of 4 minutes while the second teller has a mean of 7 minutes. There is a single queue for customers awaiting service. Suppose at noon, 3 customers enter the system. Customer A goes to the first teller, B to the second teller, and C queues. To standardize the answers, let us assume that TA is the length of time in minutes starting from noon until Customer A departs, and similarly define TB and TC .
(a) What is the probability that Customer A will still be in service at time 12:05?
(b) What is the expected length of time that A is in the system?
(c) What is the expected length of time that A is in the system if A is still in the system at
12:05?
(d) How likely is A to finish before B?
(e) What is the mean time from noon until a customer leaves the bank?
(f) What is the average time until C starts service?
(g) What is the average time that C is in the system?
(h) What is the average time until the system is empty?
(i) What is the probability that C leaves before A given that B leaves before A?
(j) What are the probabilities that A leaves last, B leaves last, and C leaves last?
(k) Suppose D enters the system at 12:10 and A, B, and C are still there. Let WD be the time that D spends in the system. What is the mean time that D is in the system?
2. Suppose we agree to deliver an order in one day. The contract states that if we deliver the order within one day we receive $1500. However, if the order is late, we lose money proportional to the tardiness until we receive nothing if the order is two days late. The length of time for us to complete the order is exponentially distributed with mean 0.8
J. Reed
Stochastic Manufacturing and Service Systems
Fall 2005
Homework 7
November 22, 2005
Due at the start of class on Thursday, December 1st.
1. Suppose there are two tellers taking customers in a bank. Service times at a teller are independent, exponentially distributed random variables, but the first teller has a mean service time of 4 minutes while the second teller has a mean of 7 minutes. There is a single queue for customers awaiting service. Suppose at noon, 3 customers enter the system. Customer A goes to the first teller, B to the second teller, and C queues. To standardize the answers, let us assume that TA is the length of time in minutes starting from noon until Customer A departs, and similarly define TB and TC .
(a) What is the probability that Customer A will still be in service at time 12:05?
(b) What is the expected length of time that A is in the system?
(c) What is the expected length of time that A is in the system if A is still in the system at
12:05?
(d) How likely is A to finish before B?
(e) What is the mean time from noon until a customer leaves the bank?
(f) What is the average time until C starts service?
(g) What is the average time that C is in the system?
(h) What is the average time until the system is empty?
(i) What is the probability that C leaves before A given that B leaves before A?
(j) What are the probabilities that A leaves last, B leaves last, and C leaves last?
(k) Suppose D enters the system at 12:10 and A, B, and C are still there. Let WD be the time that D spends in the system. What is the mean time that D is in the system?
2. Suppose we agree to deliver an order in one day. The contract states that if we deliver the order within one day we receive $1500. However, if the order is late, we lose money proportional to the tardiness until we receive nothing if the order is two days late. The length of time for us to complete the order is exponentially distributed with mean 0.8