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Ko ç Un iversity. OPSM 405 Service Management. Class 19: Managing waiting time: Queuing Theory. Zeynep Aksin zaksin @ku.edu.tr. it takes 8 minutes to serve a customer 6 customers call per hour one customer every 10 minutes Flow Time = 8 min. 100%. 100%. 90%. 90%. 80%. 80%. 70%.

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opsm 405 service management

Koç University

OPSM 405 Service Management

Class 19:

Managing waiting time:

Queuing Theory

Zeynep Aksin

zaksin@ku.edu.tr

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

100%

100%

90%

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Telemarketing: deterministic analysis

Flow Time Distribution

Probability

Flow Time (minutes)

telemarketing with variability in arrival times activity times
In reality service times

exhibit variability

In reality arrival times

exhibit variability

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100%

90%

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60%

Probability

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Flow Time

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Probability

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Flow Time

Telemarketing with variability in arrival times + activity times
why do queues form

Call #

10

9

8

7

6

5

4

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2

1

0

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80

1

0

0

TIME

Inventory (# of calls in system)

5

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80

1

0

0

TIME

Why do queues form?
  • utilization:
    • throughput/capacity
  • variability:
    • arrival times
    • service times
    • processor availability
a measure of variability
A measure of variability
  • Needs to be unitless
  • Only variance is not enough
  • Use the coefficient of variation
  • CV= s/m
interpreting the variability measures
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
Little’s Law

Inventory I

[units]

...

...

...

...

...

Flow Time T[hrs]

or

WIP = THROUGHPUT RATE x FLOWTIME

For a queue: N=l W

slide8

A Queueing System

c

m, CVs

Arrival

Departure

l, CVa

nL tL

nS tS

what to manage in such a process
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
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
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
Variation characteristics
  • distribution type M: CVa= CVs =1
  • distribution type D: CVa= CVs = 0
  • distribution type G: could be any value
basic notation
Basic notation

l = mean arrival rate (units per time period)

m = mean service rate (units per time period)

r = l/m = 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
Recall Little’s Law

Lq = l Wq

queue length = arrival rate * time in queue

the building block m m 1
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
Facts for M/M/1

r < 1

P0 = 1-r

Pn = P0rn

Ls = l /(m-l)

Ws = 1 / (m-l)

Lq = r l / (m-l)

Wq = r 1 / (m-l)

for a general system with c servers
For a general system with c servers

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

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

Wq = Lq / l

r=l/cm

Note:

generalized throughput delay curve

Average

Flow

Time Ws

Variability

Increases

1/m

100%

r

Utilization (ρ)

Generalized Throughput-Delay Curve
in words
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
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
slide22

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
slide23

Current Situation

2 requests/hour

Accounting

4 requests/hour

2 requests/hour

4 requests/hour

Finance

3 requests/hour

4 requests/hour

Marketing

2 requests/hour

4 requests/hour

Operations

slide24

Current Situation: queueing notation

l = 2 requests/hour

m= 4 requests/hour

Acc., Fin., Ops.

C2[A] = 1 (totally random arrivals)

C2[S] = 1

(assumption)

l = 3 requests/hour

m = 4 requests/hour

Marketing

C2[A] = 1 (totally random arrivals)

C2[S] = 1

(assumption)

current situation waiting times
Current Situation: waiting times

Accounting, Operations, Finance:

W = service time + Wq

W = 0.25 hrs. + 0.25 hrs

= 30 minutes

Marketing:

W = service time + Wq

W = 0.25 hrs. + 0.75 hrs

= 60 minutes

slide26

Proposal: Secretarial Pool

Accounting

2

Finance

2

16 requests/hour

3

Marketing

9 requests/hour

2

Operations

proposal secretarial pool
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)

slide28

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
Examples of pooling in business
  • Consolidating back office work
  • Call centers
  • Single line versus separate queues
capacity design using queueing models
Capacity design using queueing models
  • Criteria for design
    • waiting time
    • probability of excessive waiting
    • minimize probability of lost sales
    • maximize revenues
example bank branch
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 l=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
How many tellers and ATMs?

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

example
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….?
example1
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?
example2
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
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
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
Tips for queueing problems
  • Make sure you use rates not times for l and m
  • 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”