Chapter 12 Queuing Theory

1
Chapter 12 Queuing Theory

• Overview
• Definition
• Elements of a queuing model
• The role of the exponential distribution
• The birth-and-death process
• Queuing models based on the birth-and-death
  process (Poisson queuing models)
• Other queuing models
• Definition the study of queues ?quantify ?
  waiting in lines average queue length, average
  waiting time in queue, and average facility
  utilization

2
Elements of a Queuing Model

• Description Customer(s) ?arrive?Server(s)
  choose (FIFO, LIFO, etc.)?leave
• Elements of a queuing model
• Arrival distribution
• Service time distribution
• Design of service facility
• Service discipline
• Customer population (size of the queue and
  customer population
• Human behavior

3
Elements of a Queuing Model

• Output information (under steady-state
  condition)
• Ls expected number of customers in the system
• Lq expected queuing length
• Ws expected waiting time in system
• Wq expected waiting time in the queue
• Input information (a/b/c), (d/e/f)
• a arrival distribution
• b service time distribution
• c number of parallel servers
• d service discipline
• e maximum number allowed in the system
• f size of customer population

4
Role of Exponential Distribution

• Random interarrival and service times are
  described quantitatively in queuing models by the
  exponential distribution
• In most queuing situations, the arrival of
  customers occurs in a totally random exponential
  distribution is completely random
• The exponential distribution has the
  forgetfulness or lack of memory

5
The Birth-and-Death Process

• Pure Birth Model
• Arrivals only are allowed
• The interarrival time is exponential with mean
  1/?
• Result the number of arrivals during a specific
  T is Poisson with mean ?T

6
The Birth-and-Death Process

• Pure Death Model
• Departures only are permitted. At time 0 N
  customers
• The interdeparture time is exponential with mean
  1/µ
• Result the probability pn(t) of n customers
  remaining after t time units is a truncated
  Poisson distribution

7
Queuing Models Based on the Birth-and-Death
Process Poisson Queuing Models

• Poisson Assumptions The interarrival and service
  time exponential distribution
• Generalized Poisson Queuing Model
• Assumption arrival and departure rates are state
  dependent
• Given ?n, µn arrival and departure rate if n
  customers in the system
• Determine pn steady-state probability of n
  customers in the system
• Solution
• Setup the transition rate diagram
• Solve the system of balance equations and unity
  condition of probability

8
Poisson Queuing Models Single Server System

• ? arrival rate of jobs per unit time, ?n
  ? ? service rate of jobs per unit time (if
  serving facility is kept busy), ?n ?
• (M/M/1)(GD/?/?) the steady state results
• Ws 1/(? -?)
• Wq ?/?(1 - ?) where ? ?/? traffic
  intensity
• Lq ?Wq ?2/(1- ?)
• Assumption ? lt1 finite queue

9
Example

• Automata car wash facility operates with only
  one bay. Cars arrive according to a Poisson
  distribution with a mean of 4 cars per hour and
  may wait in the facilitys parking lot if the bay
  is busy. The time for washing and cleaning a car
  is exponential, with a mean of 10 minutes. Cars
  that cannot park in the lot can wait in the
  street bordering the wash facility. This means
  that for all practical purposes, there is no
  limit on the size of the system. The manager of
  the facility wants to determine the size of the
  parking lot

10
Poisson Queuing Models

• Multiple servers systems
• Use formulae
• Use tables and charts to obtain queuing
  statistics

11
Poisson Queuing Models Multiple Servers Systems

• (M/M/c)(GD/?/?). If c servers have an
  exponential distribution with service rate ? and
  arrival rate ?. if we let ? ?/? lt 1 then we
  have
• where p0 is the probability of zero customers
  in the queue,

12
Example

• A community is served by two cab companies. Each
  company owns two cabs, and the two companies are
  known to have equal shares of market. This is
  evident by the fact that calls arrive at each
  companys dispatching office at the rate of 8 per
  hour. The average time per ride is 12 minutes.
  Calls arrive according to a Poisson distribution,
  and the ride time is exponential distribution.
  The two companies recently were bought by an
  investor who is interested in consolidating them
  into a single dispatching office to provide
  better service to customers. Analyze the new
  owners proposal

13
Poisson Queuing Models Self-Service Model

• (M/M/?)(GD/N/?)

?n ? µn nµ
14
Poisson Queuing Models Machine Servicing Model

• (M/M/R)(GD/K/K), R K find a value of R
  repairmen (servers) for K machines that minimizes
  the total expected costs, which consist of the
  cost of failure and the cost of service. If ? is
  the rate of breakdown per machine and there are n
  broken machines, given the arrival rate
• and the service rate is
• Pn is the probability of n
  machines in the system. The steady-state results

Where expected number of idle repairmen
15
Example

• Toolco operates a machine shop with a total of
  22 machines. Each machine is known to break down
  once every 2 hours, on the average. It takes an
  average of 12 minutes to complete a repair. Both
  the time between breakdowns and the repair time
  follow the exponential distribution. Toolco is
  interested in determining the number of repair
  persons needed to keep the shop running smoothly

16
Other Queuing Models

• Extension of Poison queuing models
• Non-Poison queuing models
• Queuing network
• Priority-Discipline queuing models
• Queuing Decision Models