Question

In: Statistics and Probability

Suppose we have a single server in a shop and customers arrive in the shop with...

Suppose we have a single server in a shop and customers arrive in the shop with a Poisson arrival distribution at a mean rate of λ=0.5 customers per minute. The interarrival time have an exponential distribution with the average inter-arrival time being 2 minutes. The server has an exponential service time distribution with a mean service rate of 4 customers per minute. Calculate: 1. Overall system utilization 2. Number of customers in the system 3. Number of customers in the queue 4. Average time customers spend in the system 5. Average time customer spends in the queue 6. Probability all servers are busy 7. Probability an arriving customer has to wait

Solutions

Expert Solution

1)

arrivals/time period = λ= 0.5
served/time period= μ= 4
utilization factor =      ρ                                        = λ/μ             = 0.125~12.5%

2)

average number of customers in system L = λ/(μ-λ)= 0.142857

3)

average number of customers in queue Lq = λ2/(μ(μ-λ))= 0.017857

4)

average time spend in system W                   = 1/(μ-λ)= 0.285714

5)

average time spend in queue Wq                   = λ/(μ(μ-λ))= 0.035714

6)

Probability all servers are busy=0.125

7)

Probability an arriving customer has to wait =0.125

orchestra answered 2 years ago
Next > < Previous

Related Solutions

Customers arrive for service at a rate of 50 an hour and each server can deal...
Customers arrive for service at a rate of 50 an hour and each server can deal with 25 customers an hour. There are three servers. If the customers time is valued at $20 an hour and server time costs $30 an hour, how much does the queue cost for an eight hour day?
Customers arrive for service at a rate of 50 an hour and each server can deal...
Customers arrive for service at a rate of 50 an hour and each server can deal with 25 customers an hour. There are three servers. If the customers time is valued at $20 an hour and server time costs $30 an hour, how much does the queue cost for an eight hour day?
Customers arrive in a certain shop according to an approximate Poisson process on the average of...
Customers arrive in a certain shop according to an approximate Poisson process on the average of two every 6 minutes. (a) Using the Poisson distribution calculate the probability of two or more customers arrive in a 2-minute period. (b) Consider X denote number of customers and X follows binomial distribution with parameters n= 100. Using the binomial distribution calculate the probability oftwo or more customers arrive in a 2-minute period. (c) Let Y denote the waiting time in minutes until...
Customers arrive in a certain shop according to an approximate Poissonprocess on the average of two...
Customers arrive in a certain shop according to an approximate Poissonprocess on the average of two every 6 minutes. (a) Using the Poisson distribution calculate the probability of two or more customersarrive in a 2-minute period. (b) Consider X denote number of customers and X follows binomial distribution withparametersn= 100. Using the binomial distribution calculate the probability oftwo or more customers arrive in a 2-minute period. (c) Let Y denote the waiting time in minutes until the first customer arrives....
An average of 10 cars per hour arrive at a single-server drive-in teller. Assume that the...
An average of 10 cars per hour arrive at a single-server drive-in teller. Assume that the average service time for each customer is 4 minutes, and both interarrival times and service times are exponential. What is the arrival rate per minute? What is the servicing rate per minute? What is the servicing rate per hour? What is the traffic intensity? What is the probability that the teller is idle? What is the average number of cars waiting in line for...
Customers arrive at a two-server system at a Poisson rate λ=5. An arrival finding the system...
Customers arrive at a two-server system at a Poisson rate λ=5. An arrival finding the system empty is equally likely to enter service with either server. An arrival finding one customer in the system will enter service with the idle server. An arrival finding two others will wait in line for the first free server. The capacity of the system is 3. All service times are exponential with rate µ=3, and once a customer is served by either server, he...
Customers arrive at a two-server system according to a Poisson process having rate λ = 5....
Customers arrive at a two-server system according to a Poisson process having rate λ = 5. An arrival finding server 1 free will begin service with that server. An arrival finding server 1 busy and server 2 free will enter service with server 2. An arrival finding both servers busy goes away. Once a customer is served by either server, he departs the system. The service times at server i are exponential with rates μi, where μ1 = 4, μ2...
Consider three servers. An average of 12 customers per hour arrive from outside at server 1,...
Consider three servers. An average of 12 customers per hour arrive from outside at server 1, an average of 36 customers per hour arrive from outside at server 2 and an average of 24 customers per hour arrive from outside at server 3. Interarrival times are exponential. Servers 1, 2 and 3 can serve at exponential rates of 100, 120 and 80 customers per hour respectively. After completing service at server 1, 25% of the customers leave the system and...
Customers arrive at a booking office window being manned by a single individual at a rate...
Customers arrive at a booking office window being manned by a single individual at a rate of 25 per hour. The time required to serve a customer has a mean of 120 seconds. If the space in front of the window can accommodate only 2 customers. Then the probability that a arriving customer will have to wait outside the provided space will be?
Question 3(a): When customers arrive at Cool's Ice Cream Shop, they take a number and wait...
Question 3(a): When customers arrive at Cool's Ice Cream Shop, they take a number and wait to be called to purchase ice cream from one of the counter servers. From experience in past summers, the store's staff knows that customers arrive at a rate of 150 per hour on summer days between 3:00 p.m. and 10:00 p.m., and a server can serve 1 customer in 1 minute on average. Cool's wants to make sure that customers wait no longer than...
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT