Ko ç University

OPSM 301: Operations Management

Session 20: Queue Management

Zeynep Aksin [email protected]

The Service Process

    Customer Inflow (Arrival) Rate (

R i

) (  ) – Inter-arrival Time = 1 /

R i

Processing Time

T p

–

(unit load)

Processing Rate per Server = 1/

T p

(µ) Number of Servers (

c

) – Number of customers that can be processed simultaneously Total Processing Rate (Capacity) =

R p

=

c / T p

(cµ)

Operational Performance Measures

processing (  ) R i e.g10 /hr waiting R (  ) 10 /hr

T w ?

10 min, R p =12/hr

 Flow time

T = T w + T p (waiting+process)

 Inventory

I

 Flow Rate

R =

=

I w +

Min (

R i , R p

)

I p

     Stable Process =

Little’s Law: I = R

Capacity Utilization 

R i T,

<

R p

, , so that

R

=

R i



= R i I w / R p = R < 1



T w , I p

Safety Capacity =

R p – R i

Number of Busy Servers =

I p = c



= R i



T p = R



T p

Summary: Causes of Delays and Queues



High Unsynchronized Variability

in – Interarrival Times – Processing Times 

High Capacity Utilization

Safety Capacity

R s

=

R p

 =

R i / R p ,

or Low

– R i ,

due to – – High Inflow Rate

R i

Low Processing Rate

R p

time, or few servers) =

c/ T p

(i.e. long service

The Queue Length Formula

I w

 ρ 1  2(c  1) ρ

CV i

2 

CV p

2 2

Utilization effect

x

Variability effect

where  

R i / R p ,

where

R p = c / T p ,

and

CV i

and

CV p

are the Coefficients of Variation (Standard Deviation/Mean) of the inter-arrival and processing times (assumed independent)

Average Time in System

T

Throughput- Delay Curve

Variability Increases

T p

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

Deriving Performance Measures from Queue Length Formula



Use the formula to find I w

 

T w = I w /R T = T w + T p

 

I p

=

T p R I =I w + I p

I w

How can we reduce waiting?

 ρ 1  2(c  1) ρ

CV i

2 

CV p

2 2    Reduce utilization: – Increase capacity: faster servers, better process design, more servers Reduce variability – – Arrival: Appointment system Service:Standardization of processes, automation We can control arrivals – – – Short lines (express cashiers) Specific hours for specific customers Specials (happy hour)

Example :Effect of pooling

       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 Assume inter-arrival time for requests and service times have exponential distribution (i.e. CV=1)

Current Situation

Accounting 2 requests/hour 2 requests/hour Finance 3 requests/hour Marketing Operations 2 requests/hour 4 requests/hour 4 requests/hour 4 requests/hour 4 requests/hour 11

Current Situation: waiting times

Accounting, Operations, 

I w T w

   2  Finance: 0 .

5 , c  1 4 0 .

5 2x2 1  1 

I

1 0.5

w R

 0 .

5 2 2  0 1 .

25 0 .

5 2  0 .

5  0 .

5 T =processing time+waiting time =0.25 hrs. +

0.25 hrs =0.5 hrs=30 min

Marketing:   3 4  0 .

75 , c  1

I w

 0 .

75 2x2 1 0.75

1  1  2 0 .

75 2 1  0 .

75  2 .

25

T w

 2 .

25  0 .

75 3 T =processing time+waiting time =0.25 hrs. +

0.75 hrs =1 hr=60 min

12

Proposal: Secretarial Pool

Accounting Finance Marketing 2 2 3 2 9 requests/hour Operations

Arrival rate=R=9/hr

Utilization=Ri/Rp=9/16

Tp=1/4 hr, R p =c/T p =16/hr

13

Proposed System: Secreterial pool



I w

 9 16   0 .

5625

c

2 ( 4  1 ) 0 .

5625 1  0 .

5625 1  2  4 R 1  0 .

 37 9

T w



I R w

 0 .

37 9  0 .

04 T =processing time+waiting time =0.25 hrs. +

0.04 hrs =0.29 hr=17.4 min

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).

Pooling improves waiting times by ensuring effective use of capacity

R i R i R i /2 Queue 1 R i /2 Queue 2

Effect of Pooling

Server 1 Server 2 Queue Server 1

Pooled service capacity reduces waiting

Server 2

Examples of pooling in business

   Consolidating back office work Call centers Single line versus separate queues

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

Intuition building exercise

  Check the following website: Waiting Line Simulation (use internet explorer) 

http://archive.ite.journal.informs.org/Vol7No1/DobsonShumsky/security_simulatio n.php

  Run six different examples. Suggestion (you can use different numbers): – Arrival rate=9, service rate=10 , CV=0, CV=1, CV=2 CV=0.5

– Arrival rate =9, service rate=12 CV=1 CV=0.5

write down the parameters and the average performance measures to observe the effect of utilization and variability on waiting times.

effect of variability and utilization.

Compare the simulation output with the results you find using formulas. Note the

18

Exercise: Example 1

An automated pizza vending machine heats and dispenses a slice of pizza in exactly 4 minutes. Customers arrive at a rate of one every 6 minutes with the arrival rate exhibiting a Poisson distribution.

Determine: A) The average number of customers in line.

B) The average total waiting time in the system.

R i =1/6 per min=10/hr T p =4 min, c=1 R p =15/hr =10/15=0.66

CV i =1, CV p =0

I w

 0 .

66 2 1  0 .

66 1 2  0 .

64 customers

T w T

 

T I w R w

 

T p

0 .

66  0 .

064

hr

10  7 .

84 min Exercise: 1. What if we have a human server, with CV=1?

2.What is the effect of buying a second machine?

19

   

Exercise Example 2: Computing Performance Measures

Given – Interarrival times: 10, 10, 2, 10, 1, 3, 7, 9, and 2 seconds • Avg=6, stdev=3.937,

R i =1/6

– Processing times: 7, 1, 7, 2, 8, 7, 4, 8, 5, 1 seconds • Avg=5, stdev=2.8284

–

c

= 1,

R p =1/5

Compute – Capacity Utilization

r = R i

–

CV i

= 3.937/6 = 0.6562

–

CV p

= 2.8284/5 = 0.5657

/ R p

= 5/6=0.833

Queue Length Formula –

I w

= 1.5633 Hence –

T w

=

I w

–

T

/

R

= 9.38 seconds, and = 14.38 seconds, so

T p

–

I

=

RT

= 5 seconds, so = 14.38/6 = 2.3966 customers in the system

Example 2:Effect of Increasing Capacity

    Assume an indentical server is added (c=2). Given – Interarrival times: 10, 10, 2, 10, 1, 3, 7, 9, and 2 • Avg=6, stdev=3.937,

R i =1/6

– Processing times: 7, 1, 7, 2, 8, 7, 4, 8, 5, 1 • Avg=5, stdev=2.8284

–

c

= 2,

R p =2/5

Compute – Capacity Utilization

r = R i

–

CV i

= 3.937/6 = 0.6562

–

CV p

= 2.8284/5 = 0.5657

/ R p

= 0.4167

Queue Length Formula –

I w

= 0.07536 Hence –

T w

=

I w

–

T

/

R

= 0.45216 seconds, and = 5.45216 seconds, so

T p

–

I

=

RT

= 5 seconds, so = 5.45216/6 = 0.9087 customers in the system

Capacity planning

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 TL 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 TL 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?

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 Scheduled arrivals to reduce demand variability Initiatives to smoothen arrivals

Want to reduce resource interference in your processes: how?

     smaller lotsizes (smaller batches) better balanced line  by speeding-up bottleneck (adding staff, changing procedure, different incentives, change technology)  through cross-training eliminate steps buffers integrate work (pooling)