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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)
Pro
bab
ilit
y
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0 15 30 45 60 75 90
105
120
135
150
165
180
195
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Telemarketing with variability inarrival times + activity times
In reality service times– exhibit variability
In reality arrival times– exhibit variability
0%
5%
10%
15%
20%
25%
0 10 20 30 40 50 60 70 80 90
100
110
120
130
140
150
160
170
180
190
Mor
e
Flow Time
Pro
bab
ilit
y
0%
20%
40%
60%
80%
100%
90%
0%
5%
10%
15%
20%
25%
30%
0 10 20 30 40 50 60 70 80 90 100
110
120
130
140
150
160
170
180
190
Mor
e
Flow Time
Pro
bab
ilit
y
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Why do queues form?
utilization: – throughput/capacity
variability: – arrival times– service times– processor availability
0123456789
10
0 20 40 60 80 100 TIME
0
1
2
3
4 5
0 20 40 60 80 100
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]
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
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
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:
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”