Queuing%20Systems

1
Queuing Systems

• Chapter 17

2
Chapter Topics

• Elements of Waiting Line Analysis
• Single-Server Waiting Line System
• Undefined and Constant Service Times
• Finite Queue Length
• Finite Calling Problem
• Multiple-Server Waiting Line
• Addition Types of Queuing Systems

3
Overview

• People and products spent significant amount of
  time in waiting lines
• Providing quick service is an important aspect of
  customer service
• Trade-off between the cost of improving service
  and the costs associated with making customers
  wait
• A probabilistic form of analysis
• Results are referred to as operating
  characteristics
• Results are used to make decisions

4
Elements of Waiting Line Analysis

• Waiting lines form because people arrive at a
  service faster than they can be served
• Not arrive at a constant rate nor are they served
  in an equal amount of time
• Have an average rate of customer arrivals and an
  average service time
• Decisions are based on these averages for
  customer arrivals and service times
• Formulas are used to compute operating
  characteristics

5
Single-Server Waiting Line System

• Components include
• Arrivals (customers), servers, (cash
  register/operator), customers in line form a
  waiting line
• Factors to consider
• queue discipline.
• nature of the calling population
• arrival rate
• service rate.

6
Component Definitions

• Queue Discipline order in which customers are
  served (FIFO, LIFO, or randomly)
• Calling Population source of customers
  (infinite or finite)
• Arrival Rate Frequency at which customers
  arrive at a waiting line according to a
  probability distribution (Poisson distribution)
• Service Rate Average number of customers that
  can be served during a time period (negative
  exponential distribution)
  

7
Single-Server Waiting Line System Single-Server
Model

• Assumptions
• An infinite calling population
• A first-come, first-served queue discipline
• Poisson arrival rate
• Exponential service times
• Symbology
• ? the arrival rate (average number of
  arrivals/time period)
• ? the service rate (average number served/time
  period)
• Customers must be served faster than they arrive
  (? lt ?)

8
Single-Server Queuing Formulas
Probability that no customers are in the waiting
line Probability that n customers are in the
waiting line Average number of customers in
system and waiting
line
9
SSingle-Server Queuing Formulas
Average time customer spends waiting and being
served Average time customer spends waiting in
the queue Probability that server is busy
(utilization factor) Probability that server is
idle
10
Characteristics for Fast Shop Market
? 24 customers per hour arrive at checkout
counter ? 30 customers per hour can be checked
out
11
Characteristics for Fast Shop Market
12
Steady-State Operating Characteristics

• Utilization factor, U, must be less than one
• U lt 1,or ? / ? lt 1 and ? lt ?.
• Ratio of the arrival rate to the service rate
  must be less than one or, the service rate must
  be greater than the arrival rate
• Server must be able to serve customers faster
  than the arrival rate in the long run, or waiting
  line will grow to infinite size.

13
Single-Server Waiting Line System Effect of
Operating Characteristics (1 of 6)

• Wish to test several alternatives for reducing
  customer waiting time
• Addition of another employee to pack up
  purchases
• Addition of another checkout counter.
• Alternative 1 Addition of an employee raises
  service rate from ? 30 to ? 40 customers
• Cost 150 per week, avoids loss of 75 per week
  for each minute of reduced customer waiting
  time
• System operating characteristics with new
  parameters
• Po .40 probability of no customers in the
  system
• L 1.5 customers on the average in the queuing
  system

14
Effect of Operating Characteristics

• System operating characteristics with new
  parameters (continued)
• Lq 0.90 customer on the average in the
  waiting line
• W 0.063 hour average time in the system per
  customer
• Wq 0.038 hour average time in the waiting
  line per customer
• U .60 probability that server is busy and
  customer must wait
• I .40 probability that server is available
• Average customer waiting time reduced from 8 to
  2.25 minutes worth 431.25 (8-2.25)(75) per
  week.

15
Effect of Operating Characteristics

• Alternative 2 Addition of a new checkout counter
  (6,000 plus 200 per week for additional
  cashier).
• ? 24/2 12 customers per hour per checkout
  counter
• ? 30 customers per hour at each counter
• System operating characteristics with new
  parameters
• Po .60 probability of no customers in the
  system
• L 0.67 customer in the queuing system
• Lq 0.27 customer in the waiting line
• W 0.055 hour per customer in the system
• Wq 0.022 hour per customer in the waiting
  line
• U .40 probability that a customer must wait
• I .60 probability that server is idle

16
Effect of Operating Characteristics
Savings from reduced waiting time worth 500
(8-1.33)(75) per week - 200 300 net savings
per week. After 6,000 recovered, alternative
2 would provide 300 -281.25 18.75 more
savings per week.
17
Undefined and Constant Service Times

• Constant occurs with machinery and automated
  equipment
• Constant service times are a special case of the
  single-server model with undefined service times
• Queuing formulas

18
Undefined Service Times Example

• Data Arrival rate of 20 customers per hour
  (Poisson distributed) undefined service time
  with mean of 2 minutes, standard deviation of 4
  minutes.
• Operating characteristics

19
Undefined Service Times Example (2 of 2)

• Operating characteristics (continued)

20
Constant Service Times Formulas

• No variability in service times ? 0.
• Substituting ? 0 into equations
• All remaining formulas are the same

21
Constant Service Times Example

• Inspecting one car at a time constant service
  time of 4.5 minutes arrival rate of customers of
  10 per hour (Poisson distributed).
• Determine average length of waiting line and
  average waiting time
• ? 10 cars per hour, ? 60/4.5 13.3 cars per
  hour

22
Finite Queue Length

• Length of the queue is limited.
• Operating characteristics
• M is the maximum number in the system

23
Finite Queue Length Example

• Limited parking space (one vehicle in service and
  three waiting for service)
• Mean time between arrivals of customers is 3
  minutes and mean service time is 2 minutes (both
  inter-arrival times and service times are
  exponentially distributed)
• Maximum number of vehicles in the system equals
  4.
• Operating characteristics for ? 20, ? 30, M
  4

24
Example-Cont.

• Average queue lengths and waiting times

25
Finite Calling Population

• There is a limited number of potential customers
  that can call on the system.
• Operating characteristics (Poisson arrival and
  exponential service times)

26
Finite Calling Population Example

• 20 machines each machine operates an average of
  200 hours before breaking down average time to
  repair is 3.6 hours breakdown rate is Poisson
  distributed, service time is exponentially
  distributed.
• Is repair staff sufficient?
• ? 1/200 hour .005 per hour
• ? 1/3.6 hour .2778 per hour
• N 20 machines

27
Finite Calling Population Example

• System seems inadequate.

28
Multiple-Server Waiting Line

• Two or more independent servers in parallel serve
  a single waiting line first-come, first-served
  basis
• Assumptions
• First-come first-served queue discipline
• Poisson arrivals, exponential service times
• Infinite calling population.
• Parameter definitions
• ? arrival rate (average number of arrivals per
  time period)
• ? the service rate (average number served per
  time period) per server (channel)
• c number of servers
• c ? mean effective service rate for the system
  (must exceed arrival rate)

29
Multiple-Server Waiting Line Formulas
30
Example
? 10, ? 4, c 3

31
Notation Used for Waiting Line Models

• Kendal suggested A/B/k
• A Denotes the prob. Distribution for the arrival
• B Denotes the prob. Distribution for the service
  time
• k Denotes of channels
• Letters commonly used
• M designates a Poisson prob. Dist. For the
  arrivals or an exp. prob. dist. for the service
  time
• D designates that arrivals or the service time
  is deterministic
• G designates that the arrivals or the service
  time has a general prob. Dist. With a known mean
  and variance
• M/M/1 single-server with Poisson arrivals and
  exp. service time
• M/M/2 two-server with Poisson arrivals and exp.
  service time
• M/G/1 single-server with Poisson arrivals and
  arbitrary service time

32
Problem1

• A single server queuing system with an infinite
  calling population and a FIFO discipline has the
  following arrival and service rates
• l16 customers/hour
• m24 customers/hour
• Determine P0, P3, L, Lq, W, Wq, and U