Operations Management Queuing Models - Lecture 6 (Chapter 8)

Dr. Ursula G. Kraus

1/44

Review • National Cranberry 2/44

Agenda The Effect of Variability Why do Queues form?

Performance Measures for Queuing Systems Special Queuing Models 4/44

Goldratt's Production Game Station 1 4 Station 2 4 Station 3 4 Station 5 4 Source: Eliyahu M. Goldratt, 1992, “The Goal”, North River Press, 104-112.

Station 4 4 5/44

Critical Assumptions from National Cranberry  Truck

arrivals

are constant, evenly spaced over the 11 (12) hour period  The

mix

70/30 between wet and dry berries is constant at  No variation in

processing time

processing stages at the individual 6/44

Post Office: Why Do Queues form?

• Average inter-arrival time is 4 minutes • Average processing time is 3 minutes We are processing customers faster than they arrive so what's the problem?

Source: Managing Business Process Flows (1999) 7/44

Post Office: Customer processing

Customer #

Average inter-arrival time is 4 minutes Average processing time is 3 minutes

Time Time

Source: Prof. K. Gue, Winter 03 8/44

Post Office: With constant processing time

Customer #

Average inter-arrival time is 4 minutes Average processing time is 3 minutes

Time Time

Source: Prof. K. Gue, Winter 03 9/44

Post Office: … and constant interarrival time

Customer #

Average inter-arrival time is 4 minutes Average processing time is 3 minutes

Time Time

Source: Prof. K. Gue, Winter 03 10/44

Reasons Why Queues form 

Variability:

– interarrival times – processing times – server availability

Call #

10 9 8 3 2 1 0 7 6 5 4 0 20 

(System) Utilization:

 = throughput/capacity Source: Managing Business Process Flows (1999)

Inventory (# of calls in system)

5 4 3 2 1 0 0 20 40

TIME

60 40

TIME

60 80 80 100 100 11/44

Telemarketing at L.L. Bean     During some half hours, 80% of calls dialed received a busy signal.

Customers getting through had to wait on average 10 minutes for an available agent. Extra telephone expense per day for waiting was $25,000.

For calls abandoned because of long delays, L.L.Bean still paid for the queue time connect charges.

L.L.Bean conservatively estimated that it lost $10 million of profit because of sub-optimal allocation of telemarketing resources.

Source: Managing Business Process Flows (1999). Data based on 1988 sales.

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Agenda The Effect of Variability Why do Queues form?

Performance Measures for Queuing Systems Special Queuing Models 13/44

Notation: Variability m = Mean V(X) = Variance s = Standard Deviation s

V

(

X

) 

E

 (

X



V

(

X

)  m ) 2  where X = Random Variable Source: Managing Business Process Flows (1999) 14/44

How can we

measure

variability?

Should be a

relative

measure!

Example:

standard deviation

low

s = 10 minutes variability if mean ( m ) = 4 hours

high

variability if mean ( m ) = 5 minutes

Coefficient of Variation

:

C

 s m Ratio of the standard deviation to the mean Source: Managing Business Process Flows (1999) 15/44

System Parameters   

Inputs of System

– (Average) Interarrival times:

T i

(Average) Interarrival rates:

R i

(=1/T i ) – (Average) Service times:

T p

(Average) Service rates:

R p

(=1/T p )

System structure

– Number of servers:

c

– Number of queues – Maximum queue length (buffer capacity

K)

Operating control policies

– Queue discipline (FIFO), priorities Source: Managing Business Process Flows (1999) 16/44

Process Model for Queuing Systems: Call Center Incoming calls R i R b

Waiting Queue “buffer” size K

R Queue R a Blocked Calls Abandoned Calls Server R p Source: Managing Business Process Flows (1999) Answered Calls 17/44

Queuing System: Car Wash 18/44

Notation: Throughput Rate and Capacity  (Average)

Arrival rate:

R i = 1/T i , where T i is the average interarrival time  (Average)

Throughput rate

: R = R i - R b - R a = net arrival rate  (Theoretical total)

Processing capacity:

for c servers where T p R p = c/T p is the average processing time 

System Utilization:

 = R / R p < 100% average fraction of time server is busy Source: Managing Business Process Flows (1999) 19/44

Notation: Inventory/Customers in System    Average number of customers in the

waiting

line: I q Average number of customers in

process

: I p = c  Average number of customers in the

system

(waiting and being served): I = I q + I p I

Order Queue “buffer” size K

R i I q R p I p R b R a Source: Managing Business Process Flows (1999) 20/44

Queuing System: Car Wash I q I p I 21/44

Little’s Law (I = R x T)

T = I / R

I = I q + I p

waiting; in queue in process (c

 )

being served

Source: Managing Business Process Flows (1999) 22/44

Queue Length Formula - Approximation: I q   2 ( c + 1 ) 1   ´ C

2

i + C

2

p 2

utilization effect variability effect

where C i is the coefficient of variation for the interarrival times C p is the coefficient of variation for the service times Source: Managing Business Process Flows (1999) 23/44

Agenda The Effect of Variability Why do Queues form?

Performance Measures for Queuing Systems Special Queuing Models M/M/1 Queuing Model 24/44

Conditions: Poisson Distribution Patterns that occur most frequently in queuing models are

Poisson

and

Exponential Distributions

.

The number of arrivals during a time period T yield a

Poisson Distribution

if the following conditions apply: 

Certain events are occurring at random over a continuous period of time (or interval of distance, region of area, etc).



These events occur singly (one at a time), i.e. it is not possible for two events to occur exactly simultaneously.



The events also occur independently of each other, i.e. the fact that an event has occurred (or not occurred) does not affect the chance of another event occurring.

25/44

Examples: Poisson Distributions (Processes)  The number of

web page requests

arriving at a server  The number of

telephone calls

arriving at a switchboard, or at an automatic phone-switching system  The number of

photons hitting a photodetector

, when lit by a laser source  The

number of raindrops

falling over a wide area  The

arrival of customers

in a queueing system 26/44

Poisson Distribution of Arrivals and Services If arrivals at a service facility occur on a purely random fashion (one at a time), independently of each other then    The number of arrivals during a time period T yield a

Poisson Distribution

The interarrival times yields an

Exponential Distribution

The Exponential and Poisson distributions can be derived from one another.  The mean and standard deviation of the

Exponential Distribution

are equal: s  m 27/44

M/M/1 Queue (Exponential Model) The M/M/1 is a single-server queue model, that can be used to approximate a lot of simple systems . It indicates a system where: 1.

Arrivals are a

Poisson Process

with

independently

and

exponentially

distributed interarrival times 2.

Service is a

Poisson Process

where service time is

independently

and

exponentially

distributed 3.

There is

one server

Note: “M” stands for Markovian, a reference to the memoryless property of the exponential distribution 30/44

Queue Length for an M/M/1 Queue I I ( ( + 1 ) ) i i 2 p I •Independent, exponental T i •Single server and T p

q

 1  2   Source: Managing Business Process Flows (1999) 32/44

M/M/1 Queue  Average number of customers in the

waiting

I I q q   2 line:

(exact result)

 Average number of customers in the

system

 (waiting and being served)

I



I q

+

I p

  2 1   +    1   Little’s Law: Average total time in the

system

(Flow Time)

T



I

/

R T q T p

 

I q

/

I p

/

R R

Source: Managing Business Process Flows (1999) 33/44

Utilization and Variability Drive Congestion Average Flow Time

T T



I

/

R

Variability Increases

T p

Source: Managing Business Process Flows (1999) Utilization (ρ) 100%  34/44

Example: Calling Center Consider a mail order company that has one customer service representative (CSR) taking calls. When the CSR is busy, the caller is put on hold. The calls are taken in the order received.

Each minute a caller spends on hold costs the company $2 in telephone charges, customer dissatisfaction and loss of future business. In addition, the CSR is paid $20 an hour.

Assume that calls arrive exponentially at the rate of one every 3 minutes. The CSR takes on average 2.5 minutes to complete the order. The time for service is also assumed to be exponentially distributed. (a) What is the capacity utilization?

(b) How many customers will be on average on hold?

(c) What is the average time spent on hold?

(d) Estimate the average hourly cost of operating the call center.

Source: Managing Business Process Flows (1999) 35/44

Cost of Service Tradeoff Level of service Total cost (Service) Capacity Cost (Customer) Waiting Cost 37/44

Example: Ticket Outlet Customers arrive at a suburban ticket outlet at a rate of 14 per hour on Monday mornings. Selling the tickets and providing general information takes an average of 3 minutes per customer. There is one ticket agent on duty on Mondays. Assume exponential interarrival and service times.

a) b) c) d) What percentage of time is the ticket agent busy?

How many customers are in line, on average?

How many minutes, on average, will a customer spend in the system?

What is the average waiting time in queue?

Source: Managing Business Process Flows (1999) 38/44

Performance Measures to Focus on 

Sales

– Throughput (

R

) – Abandoning rate (

R a

) 

Cost

– Capacity utilization ( ) – Queue length (

I q

) ; Total number in process (

I

) 

Customer Service

– Waiting time in queue (

T q

)

;

– Blocking rate (

R b

) Total time in process (

T)

Source: Managing Business Process Flows (1999) 41/44

Levers for System Improvements  Increase process capacity R p – adding more servers – working faster (decreasing T p )  Reduce waiting time T q and queue length I q – reduce variability – decrease arrival rate (not desirable in general) Source: Managing Business Process Flows (1999) 42/44

Example: Calling Center with 2 CSRs (2 servers) I.

– 2 phone numbers Company hires a second CSR who is assigned a new telephone number. Customers are now free to call either of the two numbers. Once they are put on hold customers tend to stay on line since the other may be worse II.

– 1 phone number: pooling Both CSRs share the same telephone number and the customers on hold are in a single queue 50% Queue Server 50% Queue Server Queue Servers Source: Managing Business Process Flows (1999) 43/44