1. VIRTUE MARYLEE
MUGURACHANI
QUEING THEORY
BIRTH and DEATH

2. Overview
Introduction to Queuing Theory
Uses and Application
Element of Queuing theory
Birth and Death of Queuing Theory

3. Queuing Theory Mathematics of waiting lines
Useful in predicting and evaluating system performance

4. Uses Operation Research Manufacturing and Systems analysis
describe the behavior of queuing system
Evaluate system performance

5. APPLICATION
Health Services
Telecommunications
Manufacturing System

6. Model of a Queuing System
Waiting time to receive services
Arrival of a customer
Departure after being served
input
output
Server
Queue

7. Element of Queuing System
Customers
Server

8. Example: “Car Wash”

9. Characteristics of Queuing System
Arrival Process
Distribution that describes a task arrive in the system
Service Process
The distribution that determines the task processing time
Number of servers
Total number of servers available to process the tasks

10. Birth and death Process
Continuous Markov Process
2 state transition
Increases state variable by one
decreases state variable by one

11. Birth and Death Process act as the foundation of commonly used queuing models
Birth: equivalent to the arrival of a customer or job
Death: equivalent to departure of a served customer or job

12. APPLICATION Demography Epidemiology Biology Queuing theory
Performance engineering

13. Assumption 1. Given N(t) = n
The time until next birth (TB ) is exponentially distributed with parameter ƛn (Customers arrive according to Poisson Process)
The remaining customer service time (TD ) is exponentially distributed with parameter µn

14. 2 TB and TD are mutually independent stochastic variables and state transition occur through exactly one birth (n n+1) or one death (n n-1)

15. ƛ

16. Birth and death Process rate diagram2 1

17. ƛ ƛ 1 2

18. EXAMPLE ƛ : 2/sec µ : 3/sec 𝑈𝑡𝑖𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛 = ƛ µ = 2 3
𝑈𝑡𝑖𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛 = ƛ µ = 2 3
What Percentage are We in State 0
P0 = 1 – Utilization
P1 = ƛ µ

19. THANK YOU