10. Customers arrive an average of 8 per hour and an average of 12 customers can be served in an hour. Assume this is an M/M/1 model. (Noteshaper Quick Start #18 - #20) what is the system utilization? what is the average length of the line? what is the average number of customers in the system? what is the average amount of time spent waiting in the line? what is the average amount of time a customer spends in the system? what is the probability of no customers in the system?
10. Customers arrive an average of 8 per hour and an average of 12 customers can be served in an hour. Assume this is an M/M/1 model. (Noteshaper Quick Start #18 - #20) what is the system utilization? what is the average length of the line? what is the average number of customers in the system? what is the average amount of time spent waiting in the line? what is the average amount of time a customer spends in the system? what is the probability of no customers in the system?
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Transcribed Image Text:**Queueing Theory Problems**
10. Customers arrive an average of 8 per hour and an average of 12 customers can be served in an hour. Assume this is an M/M/1 model.
(Note: Noteshaper Quick Start #18 - #20)
- **What is the system utilization?**
- **What is the average length of the line?**
- **What is the average number of customers in the system?**
- **What is the average amount of time spent waiting in the line?**
- **What is the average amount of time a customer spends in the system?**
- **What is the probability of no customers in the system?**
11. Customers arrive at a ferry ticket office at the rate of 18 per hour on Monday mornings. This can be described as a M/M/1 model. Selling the tickets and providing general information takes an average of 2 minutes per customer. One ticket agent is on duty on Mondays.
(Note: Noteshaper Scenario #34)
a. **What is the average length of the line on Monday mornings?**
b. **On average, how long does a customer wait to buy a ticket on Monday mornings (in minutes)?**
c. **How long does it take to successfully buy a ticket on Monday mornings (in minutes)?** (This includes time waiting in line and purchasing from the agent.)
d. **What is the probability that an arriving customer has to wait to buy a ferry ticket on Monday morning?**
e. **What is the probability of exactly four customers in the ferry ticket office?** This includes both customers waiting in line and those being served.
![# Operating Characteristics of the M/M/1 Queuing Model
## Definitions:
- **\( W \)** refers to time.
- **\( L \)** refers to the number of customers.
- **\( q \)** refers to the queue.
- **\( s \)** refers to the system.
## Parameters:
- **\( \lambda \) (Lambda)**: Average number of arrivals.
- **\( \mu \) (Mu)**: Average number served.
## System Utilization
\[
\frac{\lambda}{\mu}
\]
## Probability of No Customers in the System
\[
P_0 = 1 - \frac{\lambda}{\mu}
\]
## Probability of \( n \) Customers in the System
\[
P_n = P_0 \left(\frac{\lambda}{\mu}\right)^n
\]
## Average Number of Customers in the Queue
\[
L_q = \frac{\lambda^2}{\mu(\mu - \lambda)}
\]
## Average Number of Customers in the System
\[
L_s = L_q + \frac{\lambda}{\mu}
\]
## Average Waiting Time in the Queue
\[
W_q = \frac{L_q}{\lambda}
\]
## Average Length of Visit to the System
\[
W_s = W_q + \frac{1}{\mu}
\]
This information relates to the mathematical analysis of queues in a system where arrivals and services are stochastic, following a Poisson process for arrivals and an exponential distribution for service times. The model assumes a single server queue. It helps in evaluating the performance and efficiency of queuing systems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78a6270a-46ae-4950-aa47-fb46973de32f%2F5bcc2ead-0d13-4505-bfb3-b972a5a5cd32%2Frpqru4b_processed.png&w=3840&q=75)
Transcribed Image Text:# Operating Characteristics of the M/M/1 Queuing Model
## Definitions:
- **\( W \)** refers to time.
- **\( L \)** refers to the number of customers.
- **\( q \)** refers to the queue.
- **\( s \)** refers to the system.
## Parameters:
- **\( \lambda \) (Lambda)**: Average number of arrivals.
- **\( \mu \) (Mu)**: Average number served.
## System Utilization
\[
\frac{\lambda}{\mu}
\]
## Probability of No Customers in the System
\[
P_0 = 1 - \frac{\lambda}{\mu}
\]
## Probability of \( n \) Customers in the System
\[
P_n = P_0 \left(\frac{\lambda}{\mu}\right)^n
\]
## Average Number of Customers in the Queue
\[
L_q = \frac{\lambda^2}{\mu(\mu - \lambda)}
\]
## Average Number of Customers in the System
\[
L_s = L_q + \frac{\lambda}{\mu}
\]
## Average Waiting Time in the Queue
\[
W_q = \frac{L_q}{\lambda}
\]
## Average Length of Visit to the System
\[
W_s = W_q + \frac{1}{\mu}
\]
This information relates to the mathematical analysis of queues in a system where arrivals and services are stochastic, following a Poisson process for arrivals and an exponential distribution for service times. The model assumes a single server queue. It helps in evaluating the performance and efficiency of queuing systems.
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