queuing model summary
Download
Skip this Video
Download Presentation
Queuing Model Summary

Loading in 2 Seconds...

play fullscreen
1 / 17

Queuing Model Summary - PowerPoint PPT Presentation


  • 131 Views
  • Uploaded on

Queuing Model Summary. Assumptions of the Basic Simple Queuing Model. Arrivals are served on a first-come, first-served basis (FCFS) Arrivals are independent of preceding arrivals

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Queuing Model Summary' - dyani


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
assumptions of the basic simple queuing model
Assumptions of the Basic Simple Queuing Model
  • Arrivals are served on a first-come, first-served basis (FCFS)
  • 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 vary from one customer to another, and are independent of each other; the average service time is known
  • Service times are described by the negative exponential probability distribution
  • The service rate is greater than the arrival rate
types of queuing models a b c notation
Types of Queuing Models(A/B/C notation)
  • A: probability distribution of time between arrivals
  • B: probability distribution of service times
  • C: number of parallel servers
  • M = exponential distribution of times (or equivalent Poisson distribution of rates)
  • D = deterministic or constant time
  • G = general distribution with a mean and variance (e.g., normal, uniform, or any empirical distribution)
  • Ek = Erlang distribution with shape parameter k (if k =1, Erlang equivalent to M; if k = ∞, Erlang equivalent to D)
types of queuing models a b c notation1
Types of Queuing Models(A/B/C notation)
  • Simple (M/M/1)
    • Example: Information booth at mall, line at Starbucks
  • Multi-channel (M/M/S)
    • Example: Airline ticket counter, tellers at bank
  • Constant Service (M/D/1)
    • Example: Automated car wash
  • Limited Population
    • Example: Department with only 7 copiers to service
simple m m 1 model characteristics
Simple (M/M/1) Model Characteristics
  • Type: Single-channel, single-phase system
  • Input source: Infinite; no balks, no reneging
  • Arrival distribution: Poisson
  • Queue: Unlimited; single line
  • Queue discipline: FIFO (FCFS)
  • Service distribution: Negative exponential
  • Relationship: Independent service & arrival
  • Service rate > arrival rate
simple m m 1 model equations

=

Average number of units in the system

L

s

 - 

1

=

Average time in the system

W

s

 - 

2

=

Average number of units in the queue

L

q

 ( -  )

=

Average time waiting in

the queue

W

q

 ( -  )

=

System utilization

Simple (M/M/1) Model Equations
simple m m 1 probability equations
Probability of 0 units in system, i.e., system idle:

=

-

=

-

P

1

1

0

Probability of more than k units in system:

( )

k+1

l

=

P

n>k

Where n is the number of units in the system

Simple (M/M/1) Probability Equations
multichannel m m s model characteristics
Multichannel (M/M/S) Model Characteristics
  • Type: Multichannel system
  • Input source: Infinite; no balks, no reneging
  • Arrival distribution: Poisson
  • Queue: Unlimited; multiple lines
  • Queue discipline: FIFO (FCFS)
  • Service distribution: Negative exponential
  • Relationship: Independent service & arrival
  •  Individual server service rates > arrival rate
m m s equations
(M/M/S) Equations

Probability of zero people or units in the system:

Average number of people or units in the system:

Average time a unit spends in the system:

p 0 probability of 0 units in multiple channel system needed for other calculations
P0 = Probability of 0 Units in Multiple-Channel System(needed for other calculations)

n! = 1 x 2 x 3 x 4 x……..x (n-1) x n

n0 = 1; 0! = 1

m m s equations1
(M/M/S) Equations

Average number of people or units waiting for service:

Average time a person or unit spends in the queue

constant service rate m d 1 model characteristics
Constant Service Rate (M/D/1) Model Characteristics
  • Type: Single-channel, single-phase system
  • Input source: Infinite; no balks, no reneging
  • Arrival distribution: Poisson
  • Queue: Unlimited; single line
  • Queue discipline: FIFO (FCFS)
  • Service distribution: Constant
  • Relationship: Independent service & arrival
  • Service rate > arrival rate
m d 1 equations
Average number of people or units waiting for service:

Average time a person or unit spends in the queue

Average number of people or units in the system:

Average time a unit spends in the system:

(M/D/1) Equations
limited population model characteristics
Limited Population Model Characteristics
  • Type: Single-channel, single-phase system
  • Input source: Limited; no balks, no reneging
  • Arrival distribution: Poisson
  • Queue: Limited; single line
  • Queue discipline: FIFO (FCFS)
  • Service distribution: Negative exponential
  • Relationship: Independent service & arrival
  • Service rate > arrival rate
single channel single phase manual car wash example
Single-Channel, Single-PhaseManual Car Wash Example
  • Arrival rate  = 7.5 cars per hour
  • Service rate  = an average of10 cars per hour
  • Utilization  = / = 75%
single channel single phase automated car wash example
Single-Channel, Single-PhaseAutomated Car Wash Example
  • Arrival rate  = 7.5 cars per hour
  • Service rate  = a constant rate of10 cars per hour
  • Utilization  = / = 75%
ad