OPSM 405 Service Management

OPSM 405 Service Management

Class 19:

Managing waiting time:

Queuing Theory

Koç University

Zeynep [email protected]

Telemarketing: deterministic analysis

it takes 8 minutes to serve a customer

6 customers call per hour – one customer

every 10 minutes

Flow Time = 8 min

Flow Time Distribution

Flow Time (minutes)

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Telemarketing with variability inarrival times + activity times

In reality service times– exhibit variability

In reality arrival times– exhibit variability

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Why do queues form?

utilization: – throughput/capacity

variability: – arrival times– service times– processor availability

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TIME

Call #

Inventory (# of calls in system)

A measure of variability

Needs to be unitless Only variance is not enough Use the coefficient of variation CV= /

Interpreting the variability measures

CVi = coefficient of variation of interarrival times

i) constant or deterministic arrivals CVi = 0

ii) completely random or independent arrivals CVi =1

iii) scheduled or negatively correlated arrivals CVi < 1

iv) bursty or positively correlated arrivals CVi > 1

Little’s Law

FLOWTIME

WIP = RATE THROUGHPUT

or

WIP = THROUGHPUT RATE x FLOWTIMEFor a queue: N= W

Inventory I[units] ... ...... ......

Flow Time T [hrs]

A Queueing System

c, CVs

Arrival Departure

, CVa

nL tL

nS tS

What to manage in such a process?

Inputs– Arrival rate / distribution– Service or processing time / distribution

System structure– Number of servers c– Number of queues– Maximum queue capacity/buffer capacity K

Operating control policies – Queue-service discipline

Performance Measures

Sales– Throughput– Abandoning rate

Cost– Capacity utilization– Queue length / total number in process

Customer service– Waiting time in queue / total time in process– Probability of blocking

The A/B/C notation

A: type of distribution for interarrival times B: type of distribution for service times C: the number of parallel servers

M = exponential interarrival and service time distribution (same as Poisson arrival or service rate)

D= deterministic interarrival or service time

G= general distributions

Variation characteristics

distribution type M: CVa= CVs =1

distribution type D: CVa= CVs = 0

distribution type G: could be any value

Basic notation

= mean arrival rate (units per time period)

= mean service rate (units per time period)

=/ = utilization rate (traffic intensity)

c = number of servers (sometimes also s)

P0 = probability that there are 0 customers in the system

Pn = probability that there are n customers in the system

Ls = mean number of customers in the system (Ns)

Lq = mean number of customers in the queue (Nq)

Ws = mean time in the system

Wq = mean time in the queue

Recall Little’s Law

Lq = Wq

queue length = arrival rate * time in queue

The building block: M/M/1

An infinite or large population of customers arriving independently; no reservations

Poisson arrival rate (exponential interarrival times) single server, single queue no reneges or balking no restrictions on queue length first-come first-served (FCFS) exponential service times

Facts for M/M/1

< 1

P0 = 1-

Pn = P0 n

Ls = /()

Ws = 1 / ()

Lq =

Wq = 1 / ()

For a general system with c servers

22

2

1

1

1saq CVCVW

W (or tS) = average service time + Wq (or tq )

Average wait = (scale effect) (utilization effect) (variability effect)

Wq = Lq / c

Note:

Generalized Throughput-Delay Curve

VariabilityIncreases

AverageFlowTime Ws

Utilization (ρ) 100%

In words:

in high utilization case: small decrease in utilization yields large improvement in response time

this marginal improvement decreases as the slack in the system increases

Levers to reduce waiting and increase QoS: variability reduction + safety capacity

How to reduce system variability?

Safety Capacity = capacity carried in excess of expected demand to cover for system variability– it provides a safety net against higher than

expected arrivals or services and reduces waiting time

Excel does it all!ggs.xls G/G/s Queueing Formula Spreadsheet

Inputs: Definitions of terms:lambda 6 lambda = arrival ratemu 10 mu = service rateCa^2 1 s = number of serversCb^2 1 Ca^2 = squared coeff. of variation of arrivals

Cb^2 = squared coeff. of variation of service timesNq = average length of the queueNs = average number in the systemWq = average wait in the queueWs = average wait in the system lambda/muP(0) = probability of zero customers in the system 0.6P(delay) = probability that an arriving customer has to wait

Outputs: Intermediate Calculations:s Nq Ns Wq Ws P(delay) Utilization (l/u)^s/s! sum (l/u)^s/s!

0 1.00E+00 1.00E+001 0.900000 1.500000 0.150000 0.250000 0.600000 0.600000 6.00E-01 1.60E+002 0.059341 0.659341 0.009890 0.109890 0.138462 0.300000 1.80E-01 1.78E+003 0.006164 0.606164 0.001027 0.101027 0.024658 0.200000 3.60E-02 1.82E+004 0.000615 0.600615 0.000103 0.100103 0.003486 0.150000 5.40E-03 1.82E+005 0.000055 0.600055 0.000009 0.100009 0.000404 0.120000 6.48E-04 1.82E+006 0.000004 0.600004 0.000001 0.100001 0.000040 0.100000 6.48E-05 1.82E+007 0.000000 0.600000 0.000000 0.100000 0.000003 0.085714 5.55E-06 1.82E+008 0.000000 0.600000 0.000000 0.100000 0.000000 0.075000 4.17E-07 1.82E+009 0.000000 0.600000 0.000000 0.100000 0.000000 0.066667 2.78E-08 1.82E+00

10 0.000000 0.600000 0.000000 0.100000 0.000000 0.060000 1.67E-09 1.82E+0011 0.000000 0.600000 0.000000 0.100000 0.000000 0.054545 9.09E-11 1.82E+0012 0.000000 0.600000 0.000000 0.100000 0.000000 0.050000 4.54E-12 1.82E+0013 0.000000 0.600000 0.000000 0.100000 0.000000 0.046154 2.10E-13 1.82E+00

Example: Secretarial Pool 4 Departments and 4 Departmental secretaries Request rate for Operations, Accounting, and

Finance is 2 requests/hour Request rate for Marketing is 3 requests/hour Secretaries can handle 4 requests per hour Marketing department is complaining about the

response time of the secretaries. They demand 30 min. response time.

College is considering two options:– Hire a new secretary– Reorganize the secretarial support

Current Situation

Accounting

Finance

Marketing

Operations

2 requests/hour

2 requests/hour

3 requests/hour

2 requests/hour

4 requests/hour

4 requests/hour

4 requests/hour

4 requests/hour

Current Situation: queueing notation

Acc., Fin., Ops.

Marketing

= 2 requests/hour

= 3 requests/hour

= 4 requests/hour

= 4 requests/hour

C2[A] = 1 (totally random arrivals)

C2[A] = 1 (totally random arrivals)

C2[S] = 1 (assumption)

C2[S] = 1 (assumption)

Current Situation: waiting times

W = service time + Wq

W = 0.25 hrs. + 0.25 hrs = 30 minutes

Accounting, Operations, Finance:

Marketing:

W = service time + Wq

W = 0.25 hrs. + 0.75 hrs = 60 minutes

Proposal: Secretarial Pool

Accounting

Finance

Marketing

Operations

16 requests/hour

9 requests/hour

2

2

3

2

Proposal: Secretarial Pool

Wq = 0.0411 hrs.

W= 0.0411 hrs. + 0.25 hrs.= 17 minutes

In the proposed system, faculty members in all departments get their requests back in 17 minutes on the average. (Around 50% improvement for Acc, Fin, and Ops and 75% improvement for Marketing)

The impact of task integration (pooling)

balances utilization... reduces resource interference... ...therefore reduces the impact of temporary

bottlenecks there is more benefit from pooling in a high utilization

and high variability process pooling is beneficial as long as

• it does not introduce excessive variability in a low variability system

• the benefits exceed the task time reductions due to specialization

Examples of pooling in business

Consolidating back office work Call centers Single line versus separate queues

Capacity design using queueing models

Criteria for design• waiting time• probability of excessive waiting• minimize probability of lost sales• maximize revenues

Example: bank branch

48 customers arrive per hour, 50 % for teller service and 50 % for ATM service

On average, 5 minutes to service each request or 12 per hour.

Can model as two independent queues in parallel, each with mean arrival rate of =24 customers per hour

Want to find number of tellers and ATMs to ensure customers will find an available teller or ATM at least 95 % of the time

How many tellers and ATMs?

Outputs:s Nq Ns Wq Ws P(delay) Utilization

01 infinity infinity infinity infinity 1.000000 1.0000002 infinity infinity infinity infinity 1.000000 1.0000003 0.888889 2.888889 0.037037 0.120370 0.444444 0.6666674 0.173913 2.173913 0.007246 0.090580 0.173913 0.5000005 0.039801 2.039801 0.001658 0.084992 0.059701 0.4000006 0.009009 2.009009 0.000375 0.083709 0.018018 0.333333

P(delay) or P(wait) less than 5%: 6 Tellers and 6 ATMs

Example

A mail order company has one department for taking customer orders and another for handling complaints. Currently each has a separate phone number. Each department has 7 phone lines. Calls arrive at an average rate of 1 per minute and are served at 1.5 per minute. Management is thinking of combining the departments into a single one with a single phone number and 14 phone lines.

The proportion of callers getting a busy signal will….? Average flow experienced by customers will….?

Example

A bank would like to improve its drive-in service by reducing waiting and transaction times. Average rate of customer arrivals is 30/hour. Customers form a single queue and are served by 4 windows in a FCFS manner. Each transaction is completed in 6 minutes on average. The bank is considering to lease a high speed information retrieval and communication equipment that would cost 30 YTL per hour. The facility would reduce each teller’s transaction time to 4 minutes per customer.

a. If our manager estimates customer cost of waiting in queue to be 20 YTL per customer per hour, can she justify leasing this equipment?

b. The competitor provides service in 8 minutes on average. If the bank wants to meet this standard, should it lease the new equipment?

Example

Global airlines is revamping its check-in operations at its hub terminal. This is a single queue system where an available server takes the next passenger. Arrival rate is estimated to be 52 passengers per hour. During the check-in process, an agent confirms reservation, assigns a seat, issues a boarding pass, and weighs, labels, dispatches baggage. The entire process takes on average 3 minutes. Agents are paid 20 YTL an hour and it is estimated that Global loses 1 YTL for every minute a passenger spends waiting in line. How many agents should Global staff at its hub terminal? How many agents does it need to meet the industry norm of 3 minutes wait?

Capacity Management

First check if average capacity is enough: is there a perpetual queue? If not, increase capacity

Capacity may be enough on average but badly distributed over time periods experiencing demand fluctuations: check if there is a predictable queue, do proper scheduling; you may need more people to accommodate scheduling constraints

Find sources of variability and try to reduce them: these create the stochastic queue

Want to eliminate as much variability as possible from your processes: how?

specialization in tasks can reduce task time variability standardization of offer can reduce job type variability automation of certain tasks IT support: templates, prompts, etc. incentives

Tips for queueing problems

Make sure you use rates not times forand Use consistent units: minutes, hours, etc. If the problem states “constant service times” or an

“automated machine with practically constant times” this means: deterministic service so CVs=0

Check the objective:– Cost minimization?– Service level satisfaction at lowest cost?– Etc.

Read carefully to understand difference between “waiting”, “standing in line” (in queue) “in system” or “total flow time” or “providing service”