ISE 491 Fall 2009 Dr. Burtner

11

Ozcan: Chapter 14.Queuing Models and Capacity Planning

ISE 491 Fall 2009Dr. Burtner

ISE 491 Fall 2009 Dr. Burtner Ch. 14 Ozcan 22

Outline

Learning Objectives Queuing System Characteristics

Population Source Servers Arrival Patterns Service Patterns

Specific Queue Characteristics Measures of Queuing System Performance Infinite Source-Models Model Formulations

Single Server (M/M/1) Multi Server (M/M/s>1)

Capacity Planning


Learning Objectives

Describe queuing systems and their potential applications in health care services.

Describe various queuing model formulations.

Analyze measures of queuing system performance

Recognize queuing concepts and their relationship to capacity planning.


Location: Hospital Outpatient PharmacyDay & Time: Monday, 11:00 AMAverage number in line: 20Idle servers: 0Average wait time: 15 minutes

Location: Hospital Outpatient PharmacyDay & Time: Monday, 3:30 PMAverage number in line: 0Idle servers: 3Average wait time: 0 minutesHourly wage per idle pharmacist: $40

OZCAN: Figure 14.1 Queue Phenomenon


Queuing Models

Develop a mathematical approach to the analysis of waiting lines

Weigh the cost of providing a given level of service capacity (i.e., shortening wait times) against the potential costs of having customers wait

Reduce Wait Times Minimizing costs


Costs

Waiting Costs Salaries paid to employees while they wait for

service (e.g., physicians waiting for an x-ray or test result)

Cost of waiting space Loss of business due to wait

Balking customers Reneging customers

Capacity Costs-- the cost of maintaining the ability to provide service


Total cost

Cost of service capacity

Waiting line cost

Optimum capacity

Health care service capacity (servers)

Cos

tFigure 14.2 Healthcare Service Capacity and Costs


Queuing Model Characteristics

The population source Infinite Finite

The number of servers (channels) Single line or multiple line Single phase or multiple phase

Arrival and service patterns Queue discipline (order of service)


The Population Source

Finite source Customer population where potential number of

customers is limited Infinite source

Customer arrivals are unrestricted Customer arrivals exceed system capacity Exists where service is unrestricted The model used in our text


FacilityPopulation

(Patient Source)Queue

(Waiting Line)Service

Arrivals ExitServer

Ozcan: Figure 14.3 Queuing Conceptualization of Flu Inoculations


The Number of Servers

Capacity is a function of the capacity of each server and the number of servers being used

Servers are also called channels Systems can be single or multiple channel,

and consist of phases Most healthcare systems are multiple-line

multiphase systems.


Population Receptionist(server)

Arrivals

Queue-1Nurse

(server)Physician(server)

Queue-2

Exit

Ozcan: Figure 14.4 Conceptualization of a Single-line, Multi-phase System

Phase -1

Queue-3

Phase -2

Phase -3


Ozcan: Figure 14.5 Multi-Line Queuing Systems

NursesProviding

Inoculations

Physicians-Initial Evaluation

DiagnosticTesting

Physicians-Interventions

Emergency Room

Phase-1 Phase-2 Phase-3

Line-1

Line-2

Line-3

Patients

Queue

Queue-3Queue-2Queue-1

Line-1


Arrival and Service Patterns

Both arrival and service patterns are random, thus causing waiting lines

Models assume that: Customer arrival rate can be described by a

Poisson distribution Service time can be described by a negative

exponential distribution


OZCAN: Figure 14.7 Measures of Arrival Patterns

9:00pm

Inter-arrival time

10:00pm


Ozcan: Figure 14.8 Poisson Distribution

0 1 2 3 4 5 6 7 8 9 10

Patient arrivals per hour

Pro

babi

lity

.20

.05

.10

.15


Exponential and Poisson Distributions

If service time is exponential, then the service rate isPoisson. If the customer arrival rate is Poisson, then the inter-arrival time (i.e., the time between arrivals) is exponential.

Example: If a lab processes 10 customers per hour(rate), the average service time is 6 minutes. If the arrival rate is 12 per hour, then the average time between arrivals is 5 minutes.

Thus, service and arrival rates are described by the Poisson distribution and inter-arrival times and service times are described by a negative exponential distribution.


Queue Behavior

•Balking• Patients who arrive and see big lines (the

flu shot example) may change their minds and not join the queue, but go elsewhere to obtain service; this is called balking.

•Reneging• If patients join the queue and are

dissatisfied with the waiting time, they may leave the queue; this is called reneging.


Queue Discipline

•Queue discipline refers to the order in which patients are processed.•Adaptations in healthcare:

First-Come, First-Served (may be the most common practice) Most Serious First-Served (emergency department triage system) Shortest Processing Time, First Served (sometimes applied in operating room scheduling)


System Metrics

Average number of patients waiting (in line or in system)

Average time patients wait (in line or in system)

System utilization (% of capacity utilized)

Implied costs of a given level of capacity and its related waiting line

Probability that an arriving patient will have to wait for service


0 100%System UtilizationAvg.

Num

ber o

r tim

e wa

iting

in li

ne

Increases in system utilization are achieved at the expense of increases in both the length of the waiting line and the average waiting time.

Most models assume that the system is in a steady state where average arrival and service rates are stable.

Tradeoffs


Ozcan: Exhibit 14.1 Queuing Model Classification

A: specification of arrival process, measured by inter-arrival time or arrival rate. M: negative exponential or Poisson distributionD: constant valueK: Erlang distributionG: a general distribution with known mean and variance

B: specification of service process, measured by inter-service time or service rateM: negative exponential or Poisson distributionD: constant valueK: Erlang distributionG: a general distribution with known mean and variance

C: specification of number of servers -- “s”

D: specification of queue or the maximum numbers allowed in a queuing system

E: specification of customer population


Examples of Typical Infinite-Source Models

•Single channel, M/M/s

•Multiple channel, M/M/s>1,where “s” designates the number of channels (servers).

•These models assume steady state conditions and a Poisson arrival rate.


OZCAN: Exhibit 14.2 Queuing Model Notation

arrival ratearrival rate service rateservice rateLLqq average number of customers waiting for service average number of customers waiting for serviceL L average number of customers in the system average number of customers in the system (waiting or being served)(waiting or being served)WWqq average time customers wait in line average time customers wait in lineW average time customers spend in the systemW average time customers spend in the system system utilizationsystem utilization1/1/ service time service timePPoo probability of zero units in system probability of zero units in systemPPnn probability of n units in system probability of n units in system


Infinite Source Models: Model FormulationsFive key relationships that provide basis for queuing formulations and are common for all infinite-source models:

1.The average number of patients being served is the ratio of arrival to service rate.

r

2.The average number of patients in the system is the average number in line plus the average number being served.

rLL q

3. The average time in line is the average number in line divided by the arrival rate.

q

q

LW

4. The average time in the system is the sum of the time in line plus the service time.

1

qWW

5. System utilization is the ratio of arrival rate to service capacity.

s


)(

2

qL

10P

Performance MeasuresSingle server or single team

n

n PP

0

Single Channel, Poisson Arrival and Exponential Service Time (M/M/1).


Example 14.1

A hospital is exploring the level of staffing needed for a booth in the local mall where they would test and provide information on diabetes. Previous experience has shown that, on average, every 15 minutes a new person approaches the booth.

A nurse can complete testing and answering questions, on average, in 12 minutes. If there is a single nurse at the booth, calculate system performance measures including the probability of idle time and of 1 or 2 persons waiting in the queue.

What happens to the utilization rate if another workstation and nurse are added to the unit?

Single Channel, Poisson Arrival and Exponential Service Time (M/M/1).


Arrival rate: λ = 1(hour) ÷ 15 = 60(minutes) ÷ 15 = 4 persons per hour.Service rate: μ = 1(hour) ÷ 12 = 60(minutes) ÷ 12 = 5 persons per hour.

8.54

rAverage persons served at any given time

2.3)45(5

42

qL

Number of persons in the system 48.2.3

qLL

488.042.3

q

q

LW Minutes of waiting time in the queue

601248560481

qWW Minutes in the system

(waiting and service)

Persons waiting in the queue

Example 14.1 Solution: 1


2.8.154110

P

16.)8)(.2(.)8)(.2(.54)2(. 1

11

01

PP

128.)64)(.2(.)8)(.2(.54)2(. 2

22

02

PP

Calculate queue lengths of 0, 1, and 2 persons Arrival rate: λ = 1(hour) ÷ 15 = 60(minutes) ÷ 15 = 4 persons per hour.Service rate: μ = 1(hour) ÷ 12 = 60(minutes) ÷ 12 = 5 persons per hour.




Current system utilization (s=1)

%.805*1

4

s

System utilization with an additional nurse (s=2)

%.405*2

4

s

System utilization decreases as we add more resources to it.


FIG.14.10 EXCEL SETUP AND SOLUTION TO THE DIABETES INFORMATION BOOTH PROBLEM


Multi-Channel, Poisson Arrival and Exponential Service Time (M/M/s>1)

02!1P

ssL

s

q

1

0

0

1!!

1

s

n

sn

ssn

P



a

qw W

WP

Wa = the average time for an arrival not immediately servedPw = probability that an arrival will have to wait for service.

s

Wa1



Expanding on Example 14.1: The hospital found that among the elderly, this free service had gained popularity, and now, during week-day afternoons, arrivals occur on average every 6 minutes 40 seconds (or 6.67 minutes), making the effective arrival rate 9 per hour.

To accommodate the demand, the booth is staffed with two nurses working during weekday afternoons at the same average service rate.

What are the system performance measures for this situation?



Solution:This is an M/M/2 queuing problem. The WinQSB solution provided in Exhibit 14.5 shows the 90% utilization. It is noteworthy that now each person has to wait on average one hour before they can be served by any of the nurses.

On the basis of these results, the healthcare manager may consider expanding the booth further during those hours.


Figure 14.12 System Performance Summary for Expanded Diabetes Information Booth with M/M/2


Figure 14.13 System Performance Summary for Expanded Diabetes Information Booth with M/M/3


Figure 14.14 Capacity Analysis


Multi-Channel, Poisson Arrival and Exponential Service Time (M/M/s>1) Table 14.1 Summary Analysis for M/M/s Queue for Diabetes Information Booth

Performance Measure2 Stations 3 Stations 4 Stations

Patient arrival rate 9 9 9

Service rate 5 5 5

Overall system utilization 90% 60% 45%

L (system) 9.5 2.3 1.9

Lq 7.7 0.5 0.1

W (system) – in hours 1.05 .26 0.21

Wq – in hours 0.85 .06 0.01

Po (idle) 5.3% 14.6% 16.2%

Pw (busy) 85.3% 35.5% 12.8%

Average number of patients balked 0 0 0

Total system cost in $ per hour 790.53 294.91 302.89


Assumptions and Limitations

Steady state of arrival and service process

Independence of arrival and service is assumed

The number of servers is assumed fixed

Little ability to incorporate important exceptions to the general flow

Difficult to model all but simple processes

ISE 491 Fall 2009 Dr. Burtner

Documents

Transcript of ISE 491 Fall 2009 Dr. Burtner