On priority queues with impatient customers
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On Priority Queues with Impatient Customers:. Seminar in Operations Research 01/01/2007. Exact and Asymptotic Analysis. Luba Rozenshmidt. Advisor: Prof. Avishai Mandelbaum. Flow of the Talk. Environments with heterogeneous customers Call Centers: Overview

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On priority queues with impatient customers

On Priority Queues with Impatient Customers:

Seminar in Operations Research

01/01/2007

Exact and Asymptotic Analysis

Luba Rozenshmidt

Advisor: Prof. Avishai Mandelbaum


Flow of the talk

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


Environments with priority queues

Environments with Priority Queues

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

  • Hospitals: patients – urgent, regular, surgical, …

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

  • Supermarkets: cashiers – express, regular

  • Call Centers

Examples


Call centers priority queues with impatient customers

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).


Erlang c m m n

Erlang-C (M/M/N)

Background

N

0

1

N+1

N-1

μ

(N-1)μ

  • Arrivals : Poisson(λ)

  • Service: exp(μ)

  • Number of Servers: N

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

Erlang-C Formula


Erlang a m m n m

N

N-1

0

1

N+1

μ

(N-1)μ

Nμ+θ

Nμ+2θ

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

Background

  • Arrivals : Poisson(λ)

  • Service: exp(μ)

  • Number of Servers: N

  • Individual Patience:exp(θ)

  • Steady State always exists

  • Offered Load per server ρ=λ/Nμ

Erlang-A Formula

Note:


Asymptotics background

Asymptotics: Background

Define: = Offered Load.

  • Operational Regimes

  • ED

  • QD

  • QED

; Utilization  100%, P(Wait) ≈ 1.

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

Balance between high utilization of servers and service quality

P(Wait) ≈ α, 0 < α < 1

Erlang-C: Halfin-Whitt, 1981

Erlang-A: Garnett-Mandelbaum-Reiman, 2002


Erlang a c excursions

Erlang- A/C: Excursions

N,N-1

  • T = Avg. Busy Period

  • T = Avg. Idle Period

N

N+1

N-1

0

1

N-1,N

μ

(N-1)μ

μ

μ

N

N+1

Idle Period

Busy Period

lim

rate

rate

lim


Queues with priorities

Queues with Priorities

  • N i.i.d. servers

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

  • Type j has apriorityover type k

  • FCFS within each type queue

  • where is offered load per server

  • allocated to class k

  • Type k

  • Poisson Arrivals at rateλ

  • Exponential service at rateμ

  • Exponential Patience with rate θ

    ( Total =M/M/N(+M))

k

d

High priority interrupts lower ones

  • Preemptive Priority

  • Non-Preemptive Priority

Service interruptions not allowed


Some notation priority queues

Some Notation: Priority Queues

avg. waiting time of type k under Preemptive priority

avg. waiting time of k first typesunder Preemptive priority

Similarly:

avg. waiting time of all typesunder Preemptive priority

avg. total number of delayed customers under Preemptive priority

Similarly: Non-Preemptive


Some notation related m m n m systems

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

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

k

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

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

Similarly:


Preemptive priority

Preemptive Priority

Example: K=2

Note: does not depend on service policy

Calculation of average wait of classk, , k=1,2

1)

2)


Preemptive priority1

Expected Waiting Time – Recursion based on Little’s Law

Preemptive Priority

Step 1:

Step 2:

Step 3:

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


Non preemptive priority erlang c queues

Non-Preemptive Priority:Erlang-C Queues

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

Here

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

Explanation


Non preemptive priority erlang c queues1

Non-Preemptive Priority:Erlang-C Queues

By PASTA

Erlang-C Diagram

Avg. Queue length

(given wait) M/M/N,

Avg. Busy-Period duration

M/M/1,


Non preemptive priority erlang a queues

Non-Preemptive Priority:Erlang-A Queues

  • The Highest Priority:

(Delay probability does not depend on the service discipline)


Non preemptive priority transition rate diagram

Non-Preemptive Priority:Transition-Rate Diagram

1

1

1

N,1,0

1,0,0

2,0,0

N-1,0,0

N,0,0

N,3,0

0,0,0

Nμ+θ

N,2,0

Nμ+2θ

μ

2

2

2

Nμ+θ

θ

θ

θ

1

1

1

2

N,0,1

N,1,1

N,2,1

N,3,1

Nμ+θ

L

Nμ+2θ

2

2

Nμ+2θ

2

1

1

1

2

N,3,2

N,0,2

N,1,2

N,2,2

Nμ+θ

Nμ+2θ

1

1

1

L

1

N,1

N,2

N,3

N,0

+

Nμ+θ

Nμ+2θ

Nμ+3θ


Non preemptive priority k types

Non-Preemptive Priority: K Types

Step 1:

The Algorithm

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

Step 3:


Towards example k 2

Towards : Example K=2

Non-Preemptive

Preemptive


Many servers qed example k 2

Many Servers QEDExample K=2

QED

Assume: Type2isnot negligible:

the same convergence rate!

“QD | Wait”

QD

the same limit!


Many servers ed example k 2

Many Servers EDExample K=2

ED:

Assume: Type2isnot negligible:

the same convergence rate! (=1)

the same limit!


Qed and ed with abandonments summary of results

QED and ED with Abandonments: Summary of Results


On priority queues with impatient customers

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

Higher priorities in Erlang –A enjoy QD regime (given they wait) hence “Erlang-C” performance

  • Erlang-C

  • Erlang-A

that is

Erlang-A

Erlang-C


Additional applications time varying queues

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)


Future research

Future Research

Preemptive and Non-preemptive priority

  • 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

Time-varying arrival rates

Heavy-traffic approximations


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