Operations Research · Unit 7
Queuing Theory
Exam-focused notes for Queuing Theory (Operations Research, ORS255): what the TU syllabus asks and how it has actually been tested, with 11 solved past questions from this unit.
What this unit covers
- Definition and characteristics of queues
- Queuing disciplines and service order
- Single channel queuing model
- Operating characteristics of queuing systems
- Poisson arrival processes
- Exponential service time distributions
- Average number of customers in system and queue
- Average waiting time calculations
Average waiting time calculations
In a certain bank, customers arrive in a Poisson fashion with an average time of 20 minutes between arrivals of the customers. The service time of the bank cashier follows the exponential distribution with mean time 15 minutes. Under the assumptions of single channel queuing model, find (a) The average time spent by a customer in the queue.(a);The;average;time;spent;by;a;customer;in;the;queue.(a)Theaveragetimespentbyacustomerinthequeue.(b) The probability that there are 3 customers in the bank.(b);The;probability;that;there;are;3;customers;in;the;bank.(b)Theprobabilitythatthereare3customersinthebank.[5]
- Mean inter-arrival time = 20 minutes - Mean service time = 15 minutes - Single channel model (M/M/1) Arrival rate: $$\lambda = \frac{1}{20} = 0.05 \text{ customers/min}$$ Service rate: $$\mu = \frac{1}{15} \approx 0.06667 \text{ customers/min}$$ Traffic i...
Full solved answer →Operating characteristics of queuing systems
Describe different operation characteristics of single channel queuing model. [5]
A single channel queuing model (M/M/1) consists of one server serving customers arriving from a single queue. The key operational characteristics are: - Customers arrive at an average rate of λ per unit time - Arrivals follow a Poisson distribution - Repres...
Full solved answer →What is called a queue? Describe the operating characteristics of the single channel queuing model. [5]
A queue is a linear data structure that follows the FIFO (First-In-First-Out) principle, where elements are inserted at the rear end and removed from the front end. In the context of queuing theory, a queue refers to a waiting line of customers or entities ...
Full solved answer →A TV repairman finds that the time spent on his job has an exponential distribution with mean 30 minutes. If he repairs sets in the order in which they come and if the arrival of sets is approximately Poisson with an average rate of 10 per 8 hour day. What is his expected idle time each day? How many jobs are ahead of the set just brought in? [5]
- Service time: Exponential, mean = 30 minutes - Arrival: Poisson, average rate = 10 sets per 8-hour day - Working day = 8 hours = 480 minutes Service rate: mean service time 30 min = 0.5 hour, so $$\mu = \frac{1}{0.5} = 2 \text{ sets/hour}$$ Arrival rate: ...
Full solved answer →What is called a queue? Describe the operating characteristics of the single channel queuing model. [5]
A queue is a linear data structure that follows the FIFO (First In First Out) principle, where elements are inserted at the rear end and removed from the front end. In the context of queuing theory, a queue refers to a waiting line of customers or entities ...
Full solved answer →Average number of customers in system and queue
A bank operates a single-channel queuing system with customers arriving at a rate of 6 per hour and each customer being served at an average rate of 8 per hour. Calculate (a) the average number of customers in the system and (b) the average waiting time. λ=6 per hour\lambda = 6;per;hourλ=6perhourμ=8 per hour\mu = 8;per;hourμ=8perhour[5]
- Arrival rate: $\lambda = 6$ customers/hour - Service rate: $\mu = 8$ customers/hour - Model: M/M/1 (single server, Poisson arrivals, exponential service) - Utilization: $\rho = \lambda/\mu = 6/8 = 0.75 < 1$ (stable) $$Ls = \frac{\lambda}{\mu - \lambda} = ...
Full solved answer →Customers arrive at a bank having single counter at the rate of 25 customers per hour. Time required to serve a customer has exponential distribution and average number of customers served per hour is 30. Find the average number of customers in queue and in system as well. [5]
- Arrival rate: $\lambda = 25$ customers/hour - Service rate: $\mu = 30$ customers/hour - Single counter with exponential service time → M/M/1 model $$\rho = \frac{\lambda}{\mu} = \frac{25}{30} = \frac{5}{6} \approx 0.833$$ Since $\rho < 1$, the system is s...
Full solved answer →On the average 96 patients per 24 hours day require the emergency service in clinic. Also on the average, a patient requires 10 minutes of active attention. Assume that the facility can handle only one emergency at a time. If this situation satisfies all the conditions for apply queuing theory, find the average (expected) queue length and the waiting time for the patient to be served. [5]
- Arrivals: 96 patients per 24-hour day - Service time per patient: 10 minutes (active attention) - Single server (one emergency at a time) - M/M/1 conditions satisfied Arrival rate: $$\lambda = \frac{96}{24 \text{ hr}} = 4 \text{ patients/hour}$$ Service r...
Full solved answer →Queuing disciplines and service order
Describe different types of queuing disciplines used in serving a customer in a queue. [5]
A queuing discipline (or queue discipline) is the rule that determines the order in which customers waiting in a queue are selected for service. - Also called FCFS (First Come First Served) - Customers are served in the exact order they arrive - The custome...
Full solved answer →What is called a queue? Describe different queue disciplines. [5]
A queue is a linear data structure that follows the FIFO (First-In-First-Out) principle. Elements are inserted at one end called the rear (or tail) and removed from the other end called the front (or head). The first element added to the queue is the first ...
Full solved answer →Single channel queuing model
An airlines organization has one reservation clerk on duty in its local branch at any given time. The clerk handles information regarding passenger reservations and flight timing. Assume that the number of customers arriving during any given period is Poisson distributed with an arrival rate of eight per hour and that the reservation clerk can service a customer in six minutes on an average, with an exponentially distributed service time. (a) What is the probability that the system is busy? (b) What is the average time a customer spends in the system? [5]
- Arrival rate: $\lambda = 8$ customers/hour (Poisson arrivals) - Mean service time: $6$ minutes per customer (exponential) - Service rate: $\mu = \dfrac{60}{6} = 10$ customers/hour - Single server $\Rightarrow$ M/M/1 model --- The system is busy whenever t...
Full solved answer →Make Unit 7 stick
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