Queues / Lines

• You manage a call center which can answer an average of 20 calls an hour. Your call center gets 17.5 calls in an average hour. On average what is the time a customer spends on hold waiting for service?

• Who thinks that on average customers should not have to wait on hold since capacity is greater than demand?

• Who thinks the wait is going to average 21 minutes?

Why do we have to wait?

• Why do services (or almost any non-MTS manufacturer) have queues?– Processing time and or arrival time variance– Costs of capacity

• can we afford to always have more people than customers?

– Efficiency • the Dr.’s office- you wait at most steps but the Dr. is always

busy

Costs of Queues

• No queue:– wasted capacity

• can also be a competitive advantage (no waiting)

• Queue:– provider costs:

• lost customers• annoyed customers• space

– provider opportunities• move to other parts of system (increased sales)

– customer costs - wasted time

So what is a queue?

• A line– can you give examples that are not formally a line of

people?• note that for many services this queue is really a form of WIP

– do all queues serve one customer at a time?– can we have multiple queues with our system?

• So queues are then anyplace where we have work (customers in a service) waiting to be processed.

Waiting line models

• Often referred to as queuing theory• This may be one of the more useful things you

learn about this term - namely that you can manage and more importantly predict - how your processes will effect waiting times for customers. In other words you as a manager have a large amount of control over how long people wait. Or at least you can look at a proposed system and very quickly determine how long people will be delayed. The answers are often not intuitive !

Two ways to address this issue

• Queuing theory - certain types of lines can be described mathematically– all you need to do is figure out the right formulas and

plug in some numbers– easy - but requires assumptions that may not hold in your

situation– many systems have multiple lines that feed each other -

this situation is too complex for queuing theory• Simulation- we build a model of the system and

play with it- this is more work but also more flexible- so the model can be more accurate

Attributes of a queue

Calling population

Arrival process

Balk

Queue Configuration

Renege

Queue Discipline

Process / service

Departure

Need for future service?

YESNO

Attributes defined

• Calling population: anyone who might use the system– can be very broad to very narrow

• Arrival process - frequency of arrivals (or time between)• Balk - someone who would have entered the system if the line

was shorter• Renege - someone who enters the system but leaves queue

before they are served (got sick of waiting)• Configuration - how the system is laid out• Discipline - order we process work in (FIFO) for queuing theory)• Process (server)- speed at which we do the process• Need for future service - will the customer be back?

An example• You are opening an ice cream stand that has a single

employee (you). You expect to see about 25 customers an hour. It takes you an average of 2 minutes to serve each customer. Customers are served in a FIFO manner.

• Your research suggests that if there is a line of more than 4 people that some customers will leave without buying anything. In addition, if customers have to wait more than 6 minutes to get their order filled they are not likely to come back.

• How well will this system do at satisfying customers? – What assumptions are you making to answer this question?

Arrivals• To answer these questions (and many others) we can use

queuing theory. • If we assume that people arrive independently and

randomly the Poisson probability distribution is usually the best way to describe arrivals

Describing arrivals• What is the probability of n (number of) customers arriving

in a single time period (the time period is your choice)?

P(x) =

where:

x = number of arrivals in a time period = average arrival ratee = 2.71828

T = number of time periods

!xe x

For our example = 25 an hourProbability of no customers in an hour = .00000000014 25/60 = .4167 arrivals per minuteProbability of no customers in a minute = 66%Probability of 1 customer in a minute = 27%Probability of 2 customers in a minute = 5.7%Probability of 3 customers in a minute = .008What does this tell you?There is an implicit assumption here that customers arrive

one at a time- is that a big deal?

Describing service• We can use any distribution we wish to describe processing / service -

however if we are going to make assumptions or we do not know the pattern, research suggests that the exponential distribution best describes processing times in most situations. We can easily relax this assumption.

• A key characteristic of this distribution is that there is a 63% probability of servicing time being less than the mean.

More describing service

• Given our service distribution we can calculate the likelihood of serving a given customer in a given amount of time (this is not their wait - just their servicing time)

• P (service time T) = 1 - e-t (Book reversed)– where = the mean number of units that can be served

in a time period (.5 a minute for our example)– P (service time < 1 minute) = 1-e-.5(1) = 1-.6065 = 39.35%– P (service time < 1.5 minutes) = 1- e-.5(1.5) = 1-.4724 =

52.56%

What we really care about

• Describing the system in steady state.– The previous information is useful because it tells us

the likelihood of a few events. But the real use of queuing theory is to determine the steady state characteristics of our system.

• The equations on the next slides (page 715 in book) are not optimal solutions - they are how the system will behave given the data you used (and the assumptions you used)

Describing the single channel waiting line model with Poisson arrivals and Exponential service times = the mean number of arrivals per time period

– .4167 per minute in our example = the mean number of units / customers that can

be served in a time period– .5 per minute in our example

• ρ= average utilization of system = / – .4167 / .5 = 83.34%

• Pn = probability that n customers are in the system = (1- ρ) ρn

– Probability of no customers = 16.66% (did you need to calculate this? Is it a good check?)

Probability of N units in the system

Number of units Probability

0 16.66%

1 13.8

2 11.5

3 9.5

4 7.9

5 6.5

>5 34%

More describing the system• Ls = average number of customers in service system (line

and being served) = / - – .4167/(.5-.4167) = 5.002

• Lq = average number waiting in line = 2 / ( - )

• .1736 / .04165 = 4.17 this is a long line – why?• Ws = average time in system (wait and service) = 1/ ( - )

– 1/(.5-.4167) = 12.00 minutes• Wq = average time waiting = / ( - )

– .4167 / .04165 = 10 minutes

This is why queuing theory matters so much

• In the initial problem we stated that we serve 25 customers an hour - and it took 2 minutes a customer. In other words we only needed 50 minutes of capacity to serve our customers. So why do we have such long lines?

• Variance– if the customers were uniformly distributed we would

have more than 2 minutes between each– if processing time was static we would finish before the

next person arrived– in the real world this is not very common!

Handout = the mean number of arrivals per time period

– 3 per hour for Jimbo = the mean number of units / customers that can be served

in a time period– 4 per hour for Jimbo

• ρ=3/4 = .75• P0 = (1-.75)*1 = .25• L= 3 customers in system on average• Lq = 2.25 customers waiting on average• W = 1 hour in system on average• WQ =.75 hours (45 minutes) wait on average• 3 customers waiting = 4 in system = P4 = 7.9%

Another example of a system with Poisson arrivals and Exponential service times

The Donut King is trying to decide if he should advertise on radio or in the local paper. At the present time 1 employee works at his shop, and they can serve an average of 35 customers an hour. Radio advertising will cost $500 and should bring in 25 customers an hour. A newspaper add will cost $550 and bring in 30 customers an hour. Research by the Donut Institute of America suggests that people will not wait more than 12 minutes for a donut. Additionally the research suggests that if there are more than 5 people in the shop people will not even enter the donut shop. The King is trying to decide if he can handle the extra 5 people an hour he gets with a newspaper add.

Donut King Info• Radio = 25 = 35• P0 = 28.5%• L = 2.5• Lq = 1.786• W= 6 minutes• Wq = 4.29 minutes• P5 = 5.3%

• Newspaper = 30 = 35• P0 = 14%• L = 6• Lq = 5.14• W = 12 minutes• Wq = 10.28 minutes

• P5 = 6.6%

Getting more complex

• The single line / single server system is the most basic system. Most systems are more complex than this. Many people dismisses the multiple line / multiple server example- basically saying this is a bad choice. Yet it is common in settings such as supermarkets. Lets figure out why you should generally avoid this system.

Multiple channels -single queue• Perceived fairness• No getting stuck in slow

line - or behind “slow” customer

• Privacy can be enhanced• The system is more

efficient in terms of average waiting times

• Perception of long wait• Space issues• Are all customers and

employees equal?

Multiple servers / multiple queues• Multiple queues

– can differentiate lines• express lane

– division of labor• the contractor line at

builders square– customer can “chose”

server• my wife at the

supermarket– many short lines might be

better than one long line– can get stuck in slow line

Single lines are better• A single queue serving multiple servers / channels is

generally better because wait times will be less on average (nobody gets stuck)

• However this assumes a fairly high level of homogeneity among the users of the system.

• When users are very different/ have significantly different needs they should have their own line.

• So an important underlying assumption of queuing theory is that customers are fairly homogeneous and hence a single line is better.

• Is this assumption a big deal?

Getting more complex

• Queuing theory basically requires you to plug numbers into equations

• There is no value in memorizing equations– in the real world you will do one of two things

• if you need this stuff occasionally- open the book• if you use it all the time- build a spreadsheet

• This part of the test will be open book - so you can go back and see the equations

The multi-channel system

• Since we are assuming all customers are basically the same - we are also assuming all channels are about the same. If this assumption is not valid we have 2 choices.– Simulation– Solve different sets of equations for each set of unique

customer / server combinations• See page 717 - some of the equations are more

complex but we are doing the same things

Making Jimbo faster 1 or 2 channel? • 1 server = 3 = 6• ρ=3/6 = .5• P0 = .50• L = 1 customer in system on

average• Lq = 1.5 customers waiting on

average• W = 20 minutes in system on

average• WQ = 10 minute wait on

average

• 2 servers = 3 = 4• s = number of channels = 2• P0 = .4545%

• Lq = .121 customers waiting on average

• Wq = .04 hours = 2.4 minutes• W = .29 hours = 17.4 minutes• L = .87

All from page 717

Jimbo economic analysis

• Cw = cost of waiting per unit / per time period• L = average number of units in system• Cs = cost of service per time period / per channel• K = number of channels• TC= total cost per time period =

CwL + CsK

Total cost• Total cost of the computer

system equals• Cw = $30 an hour• L = 1• Cs = $3 an hour• K = 1• TC= total cost per time

period = 33

If Jimbo is really paid 8 an hour should this be 41?

• Total cost of Ned equals• Cw = $30 an hour• L = .87• Cs = $8 an hour• K = 2• TC= total cost per time

period = 42.1

Other models

• Queuing theory is very developed- and there are equations for just about every model you can think of (with a single place to wait in the system)

• For instance the book has equations for models with a finite population

• what does this mean• If population is greater than 30 use infinite models we have

been using.

Conclusions• Students often complain that management is intuitive/

common sense. The answers provided by queuing theory are often counter intuitive. And by using common sense most of you (who did not read) got the introductory question wrong

• The answers queuing theory provide matter – managing lines / waits is a key part of customer service

• If your system is pretty straightforward / fits most of the assumptions of queuing theory this is a very powerful tool. But if your system is more complex you may need simulation.

Single server equations

• ρ= average utilization of system = / • Pn = probability that n customers are in the system =

(1- ρ) ρn

• Ls = average number of customers in service system (line and being served) = / -

• Lq = average number waiting in line =2 / ( - )

• Ws = average time in system (wait and service) = 1/ ( - )

• Wq = average time waiting = / ( - )