**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.

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Practical Management Science

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Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter2: Introduction To Spreadsheet Modeling

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In an M/M/1 queueing system, the arrival rate is 5 customers per hour and the service rate is 7 customers per hour. What is the expected number of customers in the system (L)? (Round your answer to 3 decimal places.) What is the expected waiting time in the system (W)? (Express the waiting time in hours, round your answer to 3 decimal places.) What is the expected number of customers in the queue(Lq)? (Round your answer to 3 decimal places.) What is the expected waiting time in the queue(Wq)? (Express the waiting time in hours, round your answer to 3 decimal places.)
Ali Baba‘s Car Wash Service Centre is open 6 days a week, but its busiest day isalways on Sunday. From the previous data, Ali Baba estimates that dirty carsarrive at the rate of one every two minutes. One car at a time is cleaned in thisexample of a single-channel waiting line. Assuming Poisson arrivals andexponential service times, find the following: a) Compute the average number of cars in line.
Consider a bank branch that has three distinct customer arrival patterns throughout the day, as measured by average arrival rates (below). Morning (8:30 - 11:30): arrival 1 = 47 per hour. %3D Lunch (11:30 - 1:30): arrival 2 = 70 per hour. Afternoon (1:30 - 4:00): arrival 3 = 30 per hour. Regardless of the time of day, the average time it takes for a teller to serve customers is 3.17 minutes. Because of competition with other banks in the area, management has developed an internal goal to keep the average customer wait before service to be less than 4 minutes. With that in mind, answer the following: a. During the morning period, what is the minimum number of tellers that the bank needs to hire to achieve the 4-minute service goal mentioned above? [ Select] b. During lunch, what is the minimum number of tellers that the bank needs to hire to achieve the 4 minute service goal mentioned above? [ Select ] c. In the afternoon, what is the minimum number of tellers that the bank
On average, a patient spends 5 minutes waiting and 15 minutes being treated at PSH Clinic. Given the average number of patients waiting and being treated at the clinic is 50, what is the average number of patients coming through the clinic per minute on a typical day?
The time from when a patient is discharged from North Shore Hospital to the time the discharged patient’s bed is ready to be assigned to a new patient is referred to as the bed assignment turnaround time. If the bed turnaround time is excessive, it can cause problems with patient flow and delay medical procedures throughout the hospital. This can cause long waiting times for physicians and patients thus creating customer dissatisfaction. The admissions RN has assigned a patient care associate to measure the bed turnaround time for a randomly selected bed each morning., afternoon, and evening for 30 days. Following are the bed turnaround time sample observations: Day Day Bed Turnaround Times (min) 1 1 127 135 167 144 151 2 2 140 155 122 135 119 3 3 112 128 97 118 131 4 4 223 135 154 187 150 5 5 181 155 160 172 166 6 6 103 158 145 124 149 7 7 146 135 167 150 178 8 8 104 122 115 137 129 9 9 136 158 137 148 156 10 10 145 163 106 139 170 11 11 84 146 125 98…
Customers arrive at the lone check-out desk of a certain convenience store at the average rate of 0.23 per minute. If it takes on average four minutes for the store manager to serve each customer, how many customers will on average find themselves waiting in the queue? A. 10.58 B. 0.00024 C. 3.20 D. 0.00351
Safety regulations require that the time between airplane takeoffs (on the same runway) will be at least 3 minutes. When taking off, the run time of an airplane on the runway is 33 seconds. Planes are on average waiting 6 minutes and 48 seconds for takeoff. On average there are 16 planes taking off per hour. How many planes are either on the runway or waiting to take off? Round your answer to 2 decimal places.
The office has a single line for customers waiting for the next available clerk. There are two clerks who work at the same rate. On average customers arrive every 8 minutes and the average service rate is 5 per hour for each of the two clerks. The arrival rate of customers follows a Poisson distribution, while the service time follows an exponential distribution. b.) What proportion of time are both clerks idle? c.) Counting each person being served and the people in line, on average, how many customers would be in this system?

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