On Priority Queues with Impatient Customers:

Exact and Asymptotic Analysis

Seminar in Operations Research

01/01/2007

Luba Rozenshmidt

Advisor: Prof. Avishai Mandelbaum

2

Flow of the Talk

Environments with heterogeneous customers

Call Centers: Overview

Background – exact and asymptotic results

Erlang-C with priorities

Erlang-A with priorities

Asymptotic results: the lowest priority

Asymptotic results: other priorities

Additional results and future research

3

Environments with Priority Queues

Hospitals: patients – urgent, regular, surgical, …

Banks: customers – private, organizations, Platinum, Gold …

Supermarkets: cashiers – express, regular

Call Centers

Customers differ by their needs, spoken languages, potential profit, urgency ...

Examples

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Call Centers: Priority Queues with Impatient Customers

• Call centers are the primary contact channel between service providers

and their customers

U.S. Statistics

• Over 60% of annual business volume via the telephone

• 70,000 – 200,000 call centers

• 3 – 6.5 million employees (3% – 6% workforce)

• 20% annual growth rate

• $100 – $300 billion annual expenditures

• 1000’s agents in a “single" call center (large systems)

• Human aspects (impatience, abandonment).

5

Erlang-C (M/M/N)

Background

Nμ

(N-1)μ

0 1 N-1 N

μ 2μ Nμ

N+1

Nμ

•Arrivals : Poisson(λ)

•Service: exp(μ)

•Number of Servers: N

•Utilization ρ (=λ/Nμ) <1 Steady State

11

2,0

/ / /0

! 1 ! ! 1

N i NN

q Ni

P W EN i N

Erlang-C Formula

6

Erlang-A (M/M/N+M)

Background

Nμ+θ

(N-1)μ

0 1 N-1 N

μ 2μ Nμ

N+1

Nμ+2θ

•Arrivals : Poisson(λ)

•Service: exp(μ)

•Number of Servers: N

•Individual Patience: exp(θ)

Erlang-A Formula

0q ii N

P W

•Steady State always exists•Offered Load per server ρ=λ/Nμ

qP Aband E W

Note:

7

• ED

• QD

• QED

Asymptotics:

Background

Operational Regimes

, 0N R R ; Utilization 100%, P(Wait) ≈ 1.

, 0 ;N R R Short waiting time for agents, P(Wait) ≈ 0.

, ;N R R

Balance between high utilization of servers and service quality

P(Wait) ≈ α, 0 < α < 1

Define: = Offered Load.R

N

Erlang-C: Halfin-Whitt, 1981

Erlang-A: Garnett-Mandelbaum-Reiman, 2002

8

Erlang- A/C: Excursions

T = Avg. Busy Period

T = Avg. Idle Period μ

(N-1)μ

0 1 N-1 N

μ 2μ Nμ

N+1

μN N+1

Idle Period Busy Period

N,N-1

N-1,N

1

, 1 1,

, 1 1, , 1

0 1N N N Nq

N N N N N N

T TP W

T T T

, 1

1,

1

0

1

1

N N

N N

TN

TN

BusyIdle

QED00

QD0

ED0

1 N 1 N

lim limrate rate

1 N

1 N 1

N

1

N

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Queues with Priorities

• N i.i.d. servers

• K customer types, indexed k = 1, 2, …, K

• Type j has a priority over type k

• FCFS within each type queue

where is offered load per server allocated to class k

Type k

Poisson Arrivals at rate λExponential service at rate μExponential Patience with rate θ

( Total = M/M/N(+M))1 2 ... K

kk N

j k

k

d

Preemptive Priority

Non-Preemptive Priority

High priority interrupts lower ones

Service interruptions not allowed

10

Some Notation: Priority Queues

kpr qE W

1 kpr qE W

avg. waiting time of type k under Preemptive priority

avg. waiting time of k first types under Preemptive priority

1 1, , 0 , 0k k k kpr q pr q pr q pr qE L E L P W P W

pr q

pr q

E W

E L

avg. waiting time of all types under Preemptive priority

avg. total number of delayed customers under Preemptive priority

Similarly:

Similarly: Non-Preemptive

1, , 0 ,...k k knp q np q np qE W E W P W

11

Some Notation: Related M/M/N(+M) Systems

k qE W

10 , 0 , 0k

kq q qP W P W P W

avg. waiting time in M/M/N (+M) with arrival rate λk

1

q

kE W avg. waiting time in M/M/N (+M) with arrival rate 1

1

k

k ii

qE W avg. waiting time in M/M/N (+M) with arrival rate 1

K

ii

Similarly: qkqq LELELE

k ,, 1

12

Preemptive PriorityExample: K=2

1 2pr q pr q pr qE L E L E L

Calculation of average wait of class k, , k=1,2 kpr qE W

pr qE LNote: does not depend on service policy

1( 1

1pr q qE W E W

2( 1 21 2 1 2pr q pr q pr qE W E W E W

11 2 12

2

q pr q

pr q

E W E WE W

13

Preemptive Priority

Expected Waiting Time – Recursion based on Little’s Law

The Same Recursion for M/M/N and M/M/N+M Queues!

1 11 kk kpr q pr q pr qE L E L E L

1

1pr q qE W E W

1 1k kpr q qE W E W

Step 1:

Step 2:

Step 3:

1 11

1 1 1

kk kk pr q pr q k pr qkE W E W E W

1 11

1 1 1kk

k pr q pr qkkpr q

k

E W E WE W

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Non-Preemptive Priority:Erlang-C Queues

Kella & Yechielly (1985) proofs via model with vacations:

1

0

1 1qk

np qk k

P WE W

N

Here 1

k

k jj

N

- fraction of time spent with types 1, …, k

2,

1

11 1kn kp Nq kNEE W

Explanation

0 | 0k knp q

kn q qpP W E W W

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Non-Preemptive Priority:Erlang-C Queues

11 1

| 0 .1 1

k knp q q

k k

E W WN

Avg. Queue length(given wait) M/M/N,

Avg. Busy-Period duration

M/M/1,

2,0 0 , 1,...,knp q q NP W P W E k K By PASTA

Erlang-C Diagram

1 , ,k N 1 1,k N

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Non-Preemptive Priority:Erlang-A Queues

The Highest Priority:

(Delay probability does not depend on the service discipline)

1 1 10 | 0

0 0

knp q np q np q q

knp q q

E W P W E W W

P W P W

1 1

1 1 1| 0 | 0 | 0np q q q q qE W W E W W P Aband W

1 1 10 | 0k k knp q q q qE W P W E W W

17

Nμ+3θ

11

Nμ+2θ

Nμ+2θ

Nμ+θ

Nμ+θNμ+2θ

Nμ+θ

Nμ+2θNμ+θ

2

2

2

2

2

21

0,0,0 1,0,0 2,0,0 N-1,0,0 N,0,0 N,1,0N,2,0 N,3,0

N,1,1N,0,1

N,0,2 N,1,2

N,2,1 N,3,1

N,2,2 N,3,2

1

112

1 1 12

μ 2μNμ

θ

2θ

θ θ

2θ 2θ

Nμ+2θNμ+θN,0 N,1 N,2 N,3

1 1 1

L

L 1

+

Non-Preemptive Priority:Transition-Rate Diagram

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Non-Preemptive Priority: K Types

The Algorithm

Step 1:

Step 2: ”Merge” the first k types to a single type with

Step 3:

1 k 1 1 10 | 0k k k

np q q q qE W P W E W W

1 11

1 1 1

kk kk np q np q k np qkE W E W E W

1 11

1 1 1

kkk np q np qkk

np qk

E W E WE W

1

1 0 | 0np q q q qE W P W E W W

1 11 kk knp q np q np qE L E L E L

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Towards : Example K=2

Non-Preemptive

Preemptive

11

| 00

q

np q q

P Aband WE W P W

1

2

2

1

0| 0

| 0

q

np q q

q

P WE W P Aband W

P Aband W

N

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Many Servers QEDExample K=2

the same convergence rate!

the same limit!

QED

2lim 0N N

Assume: Type 2 is not negligible:

N

QD

“QD | Wait”

, ;N R R

21

ED: 1 , 0

1 1,

1

N R for some

and

the same convergence rate! (=1)

the same limit!

Many Servers EDExample K=2

N

2lim 0N N

Assume: Type 2 is not negligible:

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QED and ED with Abandonments: Summary of Results

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Erlang-C

Erlang-A

Many Servers , QED, ED Higher Priorities, Non-Preemptive: Erlang A = C

2,

11 1Nk

np qk k

EE W

N

1

0lim lim ,

1 1qk

np qN Nk k

P WE W k K

N

that is

lim | 0 lim | 0 ,k k k knp q q np q q

N NE W W E W W k K

Erlang-A Erlang-C

Higher priorities in Erlang –A enjoy QD regime (given they wait)

hence “Erlang-C” performance

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Additional Applications:Time-Varying Queues

Time-stable performance under time-varying arrivals

- ISA = Iterative Staffing Algorithm (Feldman Z. et. al. )

- Comparison with common practice (PSA, Lagged PSA) in four real call-centers

- Extension of ISA to priority queues

- Analysis of the effect of service-time distribution (Log-Normal in practice)

25

Future Research

Waiting-time distribution with current assumptions

Analysis of waits with different service/abandonment rates

Waiting-time distribution with different service / abandonment rates

Theoretical explanation of stationary ISA performance

The impact of the service-time distribution in the QED regime

Preemptive and Non-preemptive priority

Time-varying arrival rates

Heavy-traffic approximations