Characteristics of a Waiting Line System

1
Characteristics of a Waiting Line System
Queue
Arrivals
Servers

• Population of customers may be finite or infinite
• Pattern of arrivals may be random
• Customers may or may not be patient

• Queue length may be limited
• Customers are served in an FIFO sequence
• There may be more than one queues

• Service time may be random
• There may be more than one server

2
Examples of Waiting Line Systems
Single Channel Single Phase System
Service system
Served units
Arrivals
Queue
Service facility
Single Channel Multiphase System
Service system
Served units
Arrivals
Queue
Service facility
Service facility
3
Examples of Waiting Line Systems (contd.)
Multichannel Single Phase System
Service system
Served units
Service facility
Queue
Arrivals
Service facility
Example bank, post office
4
Examples of Waiting Line Systems (contd.)
Multichannel Multiphase System
Service system
Served units
Service facility
Service facility
Queue
Arrivals
Service facility
Service facility
Example laundromat, make to order production
system
5
The Pattern of Arrivals at the System has a
Poisson Distribution

• Arrivals are independent of each other
• Number of arrivals in every hour has a Poisson
  distribution

6
Examples of Poisson Distribution
?2
?4
7
The time taken to service each customer has a
Negative Exponential Distribution

• Service time of each customer is independent of
  the others.

8
Examples of Service Times with Negative
Exponential Distribution
0.06
Average service time 20 minutes
0.05
0.04
Average service time 1 hour
0.03
Probability
0.02
0.01
0
0
30
60
90
120
150
180
Service time (minutes)
9
Assumptions of the Basic Queuing Model

• Arrivals are served on a first come, first served
  basis
• Arrivals are independent of preceding arrivals
• Arrival rates are described by the Poisson
  probability distribution, and customers come from
  a very large population
• Service times are independent of one another
• Service times are described by the negative
  exponential probability distribution
• The service rate is greater than the arrival rate

10
Single Channel Single Phase System
Arrivals
Queue
Served units
Service facility

• Assumptions
• Arrivals have a Poisson distribution
• Service times have an exponential distribution
• This system is called an M/M/1 Queuing System.

11
Multichannel Single Phase System
Service facility
Queue
Arrivals
Served units
Service facility

• Assumptions
• Arrivals have a Poisson distribution
• Service times have an exponential distribution
• This system is called an M/M/s Queuing System
  where s is the number of servers.

12
Performance Measures in a Queuing System

• Average queue time, Wq
• Average queue length, Lq
• Average time in system, Ws
• Average number in system, Ls
• Probability of idle service facility, P0
• System utilization, ? ?/?
• Probability of more than k units
• in system, Pn gt k

13
Performance Measures for a Queuing System with a
single server