Queuing model summary
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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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Queuing Model Summary

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Queuing model summary

Queuing Model Summary


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%


Comparisons

Comparisons


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