1. LECTURE 09 QUEUEING THEORY PART3
ESI 4313 Spring 2013
Probabilistic Models in Operations Research
Chongsun Oh, Ph.D.

2. Priority Queueing Models
General Assumptions:
There are two or more categories of customers. Each category is assigned to a priority class. Customers in priority class 1 are given priority over customers in priority class 2. Priority class 2 has priority over priority class 3, etc.
After deferring to higher priority customers, the customers within each priority class are served on a first-come-fist-served basis.
Two types of priorities
Nonpreemptive priorities: Once a server has begun serving a customer, the service must be completed (even if a higher priority customer arrives). However, once service is completed, priorities are applied to select the next one to begin service.
Preemptive priorities: The lowest priority customer being served is preempted (ejected back into the queue) whenever a higher priority customer enters the queueing system.

3. Nonpreemptive Prioroties Queueing Model
Additional Assumptions
Nonpreemptive priorities are used as previously described.
For priority class i (i = 1, 2, … , n), the interarrival times of the customers in that class have an exponential distribution with a mean of 1/li.
All service times have an exponential distribution with a mean of 1/m, regardless of the priority class involved.
The queueing system can have any number of servers.
The utilization factor for the servers is
r = (l1 + l2 + … + ln) / sm

4. Results for the Nonpreemptive Priorities Model
Let Wk be the steady-state expected waiting time in they system (including service time) for a member of priority class k.
Expected service time 1/μ is the same for all priority classes

5. Single-server variation of the Nonpreemptive Priorities Model
A single server with different expected service time for different priority classes.
Let μk be the mean service rate for priority class k (k= 1, 2, …., N)
Steady-state expected waiting time in the system for a member of priority class k:

6. Preemptive Priorities Queueing Model
Additional Assumptions
Preemptive priorities are used as previously described.
For priority class i (i = 1, 2, … , n), the interarrival times of the customers in that class have an exponential distribution with a mean of 1/li.
All service times have an exponential distribution with a mean of 1/m, regardless of the priority class involved.
The queueing system has a single server.
The utilization factor for the server is
r = (l1 + l2 + … + ln) / m

7. Result for the Preemptive Priorities Model
Assumption: expected service time is the same for al priority classes
Total expected waiting time in the system (including the total service time)
For single server c=1
For c >1 , Wk can be calculated by an iterative procedure
Lk continue to satisfy the relationship:

8. County Hospital Example with Priorities
Divide patients into three categories
Critical : prompt treatment is vital for survival (10%)
Serious : early treatment is important to prevent further deterioration (30%)
Stable : treatment can be delayed w/o adverse medical consequences (60%)
Patients are treated in this order of priority
FCFS basis for patients in the same category
Treatment is interrupted by the arrival of a higher-priority classes
Average treatment time in ER does not differ greatly among these categories (μ=3)
λ1= 0.2, λ2= 0.6, and λ3= 1.2

9. Preemptive Priorities Model for County Hospital

10. Nonpreemptive Priorities Model for County Hospital

11. Priority Models: Steady-state Results
© The McGraw-Hill Companies, Inc., 2010

12. Queueing Networks
In many OR problems, an arrival has to pass through a series of queues arranged in a network structure a)
Simple
tandem queue b)
Traffic merging c)

13. Queueing Networks (cont’d)
We limit ourselves to a brief introduction
Equivalence property
Assume that a service facility with s servers and an infinite queue has a Poisson input with parameter λ and the same exponential service time distribution with parameter μ for each server (the M/M/s) model, where sμ > λ.
Then the steady-state output of this service facility is also a Poisson process with parameter λ.
Implication: the served customers will leave the service facility according to a Poisson process. If these customers must go to another service facility for further service, this second facility also will have a Poisson input. With an exponential service time distribution, this facility will hold the equivalence property, which can provide a Poisson input for a third facility, etc.

14. Infinite Queues in Series
Probability of having n customers at a given facility is given by the formula for Pn for the M/M/s model
The joint probability of n1 customers at facility 1, n2 customers at facility 2,…., then is the product of the individual probabilities
Product form solution

15. Jackson Networks
All outside arrivals to each queueing facility in the network must follow a Poisson process
All service times must be exponentially distributed
All queues must have unlimited capacity – infinite queue
When a job leaves one facility, the probability that it will go to another facility is independent of its past history and is independent of the location of any other job
In essence, a Jackson networks is a collection of connected M/M/s queues with known parameters
The characteristics of a Jackson network are the same as the system of infinite queues in series, except the customers visit the facilities in different order (and may not visit them all)
For each facility, its arriving customers come from both outside the system and the other facilities

16. Jackson Networks (cont’d)
A customer leaving facility i is routed next to facility j (j =1, 2, …, m) with probability pij or departs the system with probability
Under steady-state conditions, each facility j (j =1, 2, …, m) in a Jackson network behaves as if it were an independent M/M/s queueing system with arrival rate
(customers arriving from outside the system according to a Poisson input process with parameter aj)
© The McGraw-Hill Companies, Inc., 2010

17. Jackson Networks (cont’d)