1. Waiting Line Models
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Quantitative Methods of Management  Jan Fábry

2. {Potential Customers}
Waiting Line Models
Waiting Line System
Source
Service
{Server}
Waiting Area
Arrival Process
Exit
{Customers}
{Queue}
{Potential Customers}
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Quantitative Methods of Management  Jan Fábry

3. Doctor’s consultancy room
Waiting Line Models
Examples of Waiting Line Systems
Service System
Customer
Server
Doctor’s consultancy room
Patient
Doctor
Bank
Client
Clerk
Crossing
Car
Traffic lights
Airport
Airplane
Runway
Fire station
Fire
Emergency unit
Telephone exchange
Call
Switchboard
Service station
Petrol pump
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Quantitative Methods of Management  Jan Fábry

4. {Potential Customers}
Waiting Line Models
Waiting Line System
Source
Service
{Server}
Waiting Area
Arrival Process
Exit
{Customers}
{Queue}
{Potential Customers}
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Quantitative Methods of Management  Jan Fábry

5. {Potential Customers}
Waiting Line Models
Arrival Process
Source
Infinite – tourists
Finite – machines in factory
{Potential Customers}
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Quantitative Methods of Management  Jan Fábry

6. Waiting Line Models Arrival Process In batches – BUS of tourists
Arrivals
Individually – patients
Scheduled – trams, trains
Arrivals
Unscheduled – patients
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Quantitative Methods of Management  Jan Fábry

7. 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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Quantitative Methods of Management  Jan Fábry

8. Waiting Line Models Arrival Process 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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Quantitative Methods of Management  Jan Fábry

9. 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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Quantitative Methods of Management  Jan Fábry

10. 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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Quantitative Methods of Management  Jan Fábry

11. Waiting Line Models Service Process
Service configurations (type, number and arrangement of service facilities)
1. Single facility
Queue
Server
Exit
Arrival
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Quantitative Methods of Management  Jan Fábry

12. Waiting Line Models Service Process
2. Multiple, parallel, identical facilities (SINGLE queue)
Queue
Servers
Arrival
Exit
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Quantitative Methods of Management  Jan Fábry

13. Waiting Line Models Service Process
2. Multiple, parallel, identical facilities (MULTIPLE queue)
Queues
Servers
Arrival
Exit
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Quantitative Methods of Management  Jan Fábry

14. Waiting Line Models Service Process
3. Multiple, parallel, but not identical facilities
Queues
Servers
Arrival
Exit
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Quantitative Methods of Management  Jan Fábry

15. Waiting Line Models Service Process 4. Serial facilities Queue Server
Exit
Arrival
5. Combination of facilities
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Quantitative Methods of Management  Jan Fábry

16. Waiting Line Models Waiting Line Discipline of the queue
FCFS (First-Come, First-Served)
Service
{Server}
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Quantitative Methods of Management  Jan Fábry

17. Waiting Line Models Waiting Line Discipline of the queue
LCFS (Last-Come, First-Served)
Service
{Server}
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Quantitative Methods of Management  Jan Fábry

18. Waiting Line Models Waiting Line Discipline of the queue
PRI (PRIority system)
Service
{Server}
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Quantitative Methods of Management  Jan Fábry

19. Waiting Line Models Waiting Line Discipline of the queue
SIRO (Selection In Random Order)
Service
{Server}
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Quantitative Methods of Management  Jan Fábry

20. 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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Quantitative Methods of Management  Jan Fábry

21. 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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Quantitative Methods of Management  Jan Fábry

22. 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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Quantitative Methods of Management  Jan Fábry

23. 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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Quantitative Methods of Management  Jan Fábry

24. Waiting Line Models Classification of Waiting Line Models
Kendall’s notation
A / B / C / D / E / F
Probability distribution of interarrival time
Probability distribution of service time
Number of parallel servers
Queue discipline
Maximum length of queue
Size of customer’s source
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Quantitative Methods of Management  Jan Fábry

25. Standard Single-Server Exponential Model
Waiting Line Models
Standard Single-Server Exponential Model
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Quantitative Methods of Management  Jan Fábry

26. Waiting Line Models Standard Single-Server Exponential Model
( M / M / 1 / FCFS / ∞ / ∞ )
Queue
Server
Exit
Arrival
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Quantitative Methods of Management  Jan Fábry

27. 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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Quantitative Methods of Management  Jan Fábry

28. Waiting Line Models Standard Single-Server Exponential Model
Assumptions
Infinite source
Unlimited length of queue
Queue discipline is FCFS
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Quantitative Methods of Management  Jan Fábry

29. 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)
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Quantitative Methods of Management  Jan Fábry

30. 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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Quantitative Methods of Management  Jan Fábry

31. 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
Probability of an empty facility (server is idle)
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Quantitative Methods of Management  Jan Fábry

32. Waiting Line Models Standard Single-Server Exponential Model
Example – Grocery
Average waiting time in the system
Average waiting time in the queue
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Quantitative Methods of Management  Jan Fábry

33. Waiting Line Models Standard Single-Server Exponential Model
Example – Grocery
Average number of customers in the system
Average number of customers in the queue
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Quantitative Methods of Management  Jan Fábry

34. Waiting Line Models Standard Single-Server Exponential Model
Example – Grocery
Probability of finding exactly N customers in the system
P(0)
0.280
P(1)
0.202
P(2)
0.145
P(3)
0.105
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Quantitative Methods of Management  Jan Fábry

35. Waiting Line Models Standard Single-Server Exponential Model
Example – Grocery
Probability that N > n
P{N > 0}
0.720
P{N > 1}
0.518
P{N > 2}
0.373
P{N > 3}
0.269
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Quantitative Methods of Management  Jan Fábry

36. Waiting Line Models Standard Single-Server Exponential Model
Example – Grocery
Probability of being in the system longer than time t
P{T > 1 min}
0.890
P{T > 2 min}
0.792
P{T > 3 min}
0.705
P{T > 4 min}
0.627
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Quantitative Methods of Management  Jan Fábry

37. Computer Simulation
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Quantitative Methods of Management  Jan Fábry

38. Computer Simulation Analytical tools Solution Computer simulation
Computer simulation is a special method using computer experiments with the model of a real system
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Quantitative Methods of Management  Jan Fábry

39. 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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Quantitative Methods of Management  Jan Fábry