This presentation is the property of its rightful owner.
1 / 17

# 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

Queuing Model Summary

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 - - - - - - - - - - - - - - - - - - - - - - - - - -

### 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

### 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:

### 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%