Queuing Model Summary

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# Queuing Model Summary - PowerPoint PPT Presentation

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

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Presentation Transcript
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)
• 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 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
• 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

=

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
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
• 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

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:

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) 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
• 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
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
• 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-PhaseManual Car Wash Example
• Arrival rate  = 7.5 cars per hour
• Service rate  = an average of10 cars per hour
• Utilization  = / = 75%
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%