Waiting Line Models
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Operations Research
 Jan Fábry
Waiting Line Models
Waiting Line System
Source
Arrival
Process
Waiting
Area
{Customers}
{Queue}
{Potential Customers}
Service
Exit
{Server}
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Operations Research
 Jan Fábry
Waiting Line Models
Examples of Waiting Line Systems
Service System
Doctor’s consultancy room
Bank
Customer
Patient
Client
Server
Doctor
Clerk
Crossing
Airport
Fire station
Car
Airplane
Fire
Traffic lights
Runway
Emergency unit
Telephone exchange
Service station
Call
Car
Switchboard
Petrol pump
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Operations Research
 Jan Fábry
Waiting Line Models
Waiting Line System
Source
Arrival
Process
Waiting
Area
{Customers}
{Queue}
{Potential Customers}
Service
Exit
{Server}
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Operations Research
 Jan Fábry
Waiting Line Models
Arrival Process
Source
Infinite – tourists
Finite – machines in factory
{Potential Customers}
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Operations Research
 Jan Fábry
Waiting Line Models
Arrival Process
In batches – BUS of tourists
Arrivals
Individually – patients
Scheduled – trams, trains
Arrivals
Unscheduled – patients
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Operations Research
 Jan Fábry
Waiting Line Models
Arrival Process
Arrival
Arrival
Arrival
Time
Arrival rate – number of arrivals per time unit
(POISSON distribution)
Average arrival rate = 
– average number of arrivals per time unit
(mean of POISSON distribution)
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Operations Research
 Jan Fábry
Waiting Line Models
Arrival Process
Arrival
Arrival
Arrival
Time
Interarrival time – time period between two arrivals
(EXPONENTIAL distribution)
Average interarrival time = 1/
– average time period beetween arrivals
(mean of EXPONENTIAL distribution)
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Operations Research
 Jan Fábry
Waiting Line Models
Service Process
Service
{Server}
Service rate – number of customers served per time unit
(POISSON distribution)
Average service rate = 
– average number of customers served per time unit
(mean of POISSON distribution)
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Operations Research
 Jan Fábry
Waiting Line Models
Service Process
Service
{Server}
Service time – time customer spends at service facility
(EXPONENTIAL distribution)
Average service time = 1/
– average time customers spend at service facility
(mean of EXPONENTIAL distribution)
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Operations Research
 Jan Fábry
Waiting Line Models
Service Process
 Service configurations (type, number and arrangement
of service facilities)
1. Single facility
Arrival
Exit
Queue
Server
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Operations Research
 Jan Fábry
Waiting Line Models
Service Process
2. Multiple, parallel, identical facilities
(SINGLE queue)
Exit
Arrival
Queue
Servers
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Operations Research
 Jan Fábry
Waiting Line Models
Service Process
2. Multiple, parallel, identical facilities
(MULTIPLE queue)
Exit
Arrival
Queues
Servers
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Operations Research
 Jan Fábry
Waiting Line Models
Service Process
3. Multiple, parallel, but not identical facilities
Exit
Arrival
Queues
Servers
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Operations Research
 Jan Fábry
Waiting Line Models
Service Process
4. Serial facilities
Arrival
Exit
Queue
Queue
Server
Queue
Server
Server
5. Combination of facilities
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Operations Research
 Jan Fábry
Waiting Line Models
Waiting Line
 Discipline of the queue
FCFS (First-Come, First-Served)
Service
{Server}
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Operations Research
 Jan Fábry
Waiting Line Models
Waiting Line
 Discipline of the queue
LCFS (Last-Come, First-Served)
Service
{Server}
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Operations Research
 Jan Fábry
Waiting Line Models
Waiting Line
 Discipline of the queue
PRI (PRIority system)
Service
{Server}
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Operations Research
 Jan Fábry
Waiting Line Models
Waiting Line
 Discipline of the queue
SIRO (Selection In Random Order)
Service
{Server}
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Operations Research
 Jan Fábry
Waiting Line Models
Analysis of Waiting Line Models
Cost
 Waiting cost
 Service cost (facility cost)
- cost of construction
- cost of operation
- cost of maintenance and repair
- other costs (insurance, rental)
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Operations Research
 Jan Fábry
Waiting Line Models
Analysis of Waiting Line Models
Time characteristics
 Average waiting time in the queue
 Average waiting time in the system
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Operations Research
 Jan Fábry
Waiting Line Models
Analysis of Waiting Line Models
Number of customers
 Average number of customers in the queue
 Average number of customers in the system
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Operations Research
 Jan Fábry
Waiting Line Models
Analysis of Waiting Line Models
Probability characteristics
 Probability of empty service facility
 Probability of the service facility being busy
 Probability of finding N customers in the system
 Probability that N > n
 Probability of being in the system longer than time t
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Operations Research
 Jan Fábry
Waiting Line Models
Classification of Waiting Line Models
Kendall’s notation
A/B/C/D/E/F
Size of customer’s source
Maximum length of queue
Queue discipline
Number of parallel servers
Probability distribution of service time
Probability distribution of interarrival time
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Operations Research
 Jan Fábry
Waiting Line Models
Standard Single-Server
Exponential Model
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Operations Research
 Jan Fábry
Waiting Line Models
Standard Single-Server Exponential Model
( M / M / 1 / FCFS / ∞ / ∞ )
Arrival
Exit
Queue
Server
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Operations Research
 Jan Fábry
Waiting Line Models
Standard Single-Server Exponential Model
Assumptions
 Single server
 Interarrival times - exponential probability
distribution with the mean = 1/λ
 Service times - exponential probability
distribution with the mean = 1/μ
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Operations Research
 Jan Fábry
Waiting Line Models
Standard Single-Server Exponential Model
Assumptions
 Infinite source
 Unlimited length of queue
 Queue discipline is FCFS
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Operations Research
 Jan Fábry
Waiting Line Models
Standard Single-Server Exponential Model
Example – Grocery
 One shop assistant
– serves 25 customers per hour (on the average)
 From 8 a.m. to 6 p.m.
– 18 customers per hour arrive (on the average)
___________________________________________________________________________
Operations Research
 Jan Fábry
Waiting Line Models
Standard Single-Server Exponential Model
Example – Grocery
 Average arrival rate
λ = 18 customers per hour
 Average service rate
μ = 25 customers per hour
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Operations Research
 Jan Fábry
Waiting Line Models
Standard Single-Server Exponential Model
Example – Grocery
 Utilization of the system
– probability that the server is busy
– probability that there is at least 1 customer in the system



  0.72
 Probability of an empty facility (server is idle)
P(0)  1  
P(0)  0.28
___________________________________________________________________________
Operations Research
 Jan Fábry
Waiting Line Models
Standard Single-Server Exponential Model
Example – Grocery
 Average waiting time in the system
W
1
 
W  0.143 hours  8.6 minutes
 Average waiting time in the queue

Wq  W  
  (   )
1
W  0.103 hours  6.2 minutes
___________________________________________________________________________
Operations Research
 Jan Fábry
Waiting Line Models
Standard Single-Server Exponential Model
Example – Grocery
 Average number of customers in the system
L  W 

 
L  2.57 customers
 Average number of customers in the queue
2
Lq  Wq 
 (   )
Lq  1.85 customers
___________________________________________________________________________
Operations Research
 Jan Fábry
Waiting Line Models
Standard Single-Server Exponential Model
Example – Grocery
 Probability of finding exactly N customers in the system
P( N )  P(0)  N  (1   )  N
P(0)
P(1)
0.280
0.202
P(2)
P(3)
0.145
0.105
___________________________________________________________________________
Operations Research
 Jan Fábry
Waiting Line Models
Standard Single-Server Exponential Model
Example – Grocery
 Probability that N > n
PN  n   n 1
P{N > 0}
P{N > 1}
0.720
0.518
P{N > 2}
P{N > 3}
0.373
0.269
___________________________________________________________________________
Operations Research
 Jan Fábry
Waiting Line Models
Standard Single-Server Exponential Model
Example – Grocery
 Probability of being in the system longer than time t
PT  t  e (  ) t
P{T > 1 min}
P{T > 2 min}
0.890
0.792
P{T > 3 min}
P{T > 4 min}
0.705
0.627
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Operations Research
 Jan Fábry
Computer Simulation
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Operations Research
 Jan Fábry
Computer Simulation
Solution
Analytical tools
Computer simulation
Computer simulation is a special method
using computer experiments with
the model of a real system
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Operations Research
 Jan Fábry
Computer Simulation
 Entity - object that goes through the model
 Resource - agent required by the entity
 Event - significant change of the system
 Activity - process between two events
 Generating of random values
 Simulation time
 Computer simulation language
 Animation
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Operations Research
 Jan Fábry