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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

- 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

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 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).

Background

N

0

1

N+1

N-1

μ

2μ

(N-1)μ

Nμ

Nμ

Nμ

- Arrivals : Poisson(λ)
- Service: exp(μ)
- Number of Servers: N
- Utilization ρ (=λ/Nμ)<1 Steady State

Erlang-C Formula

N

N-1

0

1

N+1

μ

2μ

(N-1)μ

Nμ

Nμ+θ

Nμ+2θ

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:

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

N,N-1

- T = Avg. Busy Period
- T = Avg. Idle Period

N

N+1

N-1

0

1

N-1,N

Nμ

μ

2μ

(N-1)μ

μ

μ

N

N+1

Idle Period

Busy Period

lim

rate

rate

lim

- 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

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

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:

Example: K=2

Note: does not depend on service policy

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

1)

2)

Expected Waiting Time – Recursion based on Little’s Law

Step 1:

Step 2:

Step 3:

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

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

Here

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

Explanation

By PASTA

Erlang-C Diagram

Avg. Queue length

(given wait) M/M/N,

Avg. Busy-Period duration

M/M/1,

- The Highest Priority:

(Delay probability does not depend on the service discipline)

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

Nμ

μ

2μ

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

2θ

2θ

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

Step 1:

The Algorithm

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

Step 3:

Non-Preemptive

Preemptive

QED

Assume: Type2isnot negligible:

the same convergence rate!

“QD | Wait”

QD

the same limit!

ED:

Assume: Type2isnot negligible:

the same convergence rate! (=1)

the same limit!

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

- 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)

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