Simulating Queuing Systems

Simulation of Queuing System

Queuing System 

The analysis of waiting lines, called Queuing Theory, applies to any situation in which customers arrive at a system, wait, and receive service. Queuing Theory was developed by a Danish engineer, A. K. Erlang in 1908.



The main objectives of queuing theory are to improve customer service and reduce operating costs.

Components of the Queuing System Servicing System Servers Queue or Customer Arrivals

Waiting Line Exit

Customer Service Population Sources (Calling Source) Population Source

Finite

Infinite

Example: Number of machines needing repair when a company only has three machines.

Example: The number of people who could wait in a line for gasoline.

Customer Service Arrival Pattern (Waiting Line) Arrival Pattern

Constant

Variable

Example: A part from an automated machine arrives every 30 seconds.

Example: Customers arriving in a bank.

Waiting Line Attributes 

Input or arrival time.



Output or service rate.



Service or queue discipline. – – – –

FIFO LIFO Priority Random

Degree of Patience No Way!

No Way!

BALKING

RENEGING

Service Pattern or Service Facility Service Pattern

Constant Example: Each part takes exactly 30 seconds to make.

Variable Example: People spending time shopping.

Suggestions for Managing Queues 1.

Determine an acceptable waiting time for your customers.

3.

Try to divert your customer’s attention when waiting.

3.

Inform your customers of what to expect.

4.

Train your servers to be friendly.

5.

Encourage customers to come during the slack periods.

Single Queue Parallel Systems

Customers in a queue Servers

Working waiting room = queue

… a service completion …

An arrival …

potential customers

parallel servers

Parallel Queues Parallel Systems

Servers

Customers in parallel queues

Modeling & Simulating Queuing System A

queuing model provides measures of system performance – –

The quality of service provided to the customers The efficiency of the service operation and the cost of providing service

Modeling & Simulating Queuing System  The

quality of the service provided can be measured by – – –

Waiting time in the queue Time in the system (waiting time plus service time) Completion by a deadline

Modeling & Simulating Queuing System 

The efficiency of service operation is measured by – – – – –

Average queue length Average number of customers in the system (queue plus customers in service) Throughput – the rate at which customers are served Server utilization – percentage of time servers are busy Percentage of customers who balk or renege

Modeling & Simulating Queuing System 





The waiting time of any customer is equal to the time at which the customer begins service minus the time the customer arrived. The server is idle if the time at which customer arrives is greater than the time at which the previous customer completed service. If a customer arrives at time t, then all prior customers who have not yet completed service by time t must still be in the system.

Modeling & Simulating Queuing System 

Some observations –

–

–

–

If a customer arrives at time t and the server is not busy, then that customer can begin service immediately upon arrival. If a customer arrives at time t and the server is busy, then that customer will begin service at the time that the previous customer completes service. The time at which a customer completes service is computed as the time that the customer begins service plus the time it takes to perform the service. Once completion times are known, we may find the length of the queue.

Kendall’s Notation 

V

indicates the arrival pattern.



W

indicates the service pattern.



B

gives the number of servers.



Y

represents the system capacity.



Z

Indicates the queue discipline.

Symbols used for inter arrival time, service times & the queue disciplines Queue Characteristic

Symbol

Meaning

Inter arrival time Or Service time

D

Deterministic

M

Exponential

FIFO LIFO SIRO PRI GD

First in First out Last in First out Random order Priority ordering Any other spec-ified ordering

Queue Discipline

Cont.. 

If Y (system capacity) is not specified then set it to infinite.



If Z (queue discipline) is not specified then set it to FIFO.



M/D/2/5/FIFO (Mathematical notation) is system having: – – – – –

Exponential arrival times. Deterministic service times. Two servers. Capacity of 5 customers. FIFO queue discipline.

We welcome your queries!!

Presented by: Nitin Kapoor Richa Sharma (MCA-III)