1. On Priority Queues with Impatient Customers: Exact and Asymptotic Analysis Seminar in Operations Research 01/01/2007 Luba Rozenshmidt Advisor: Prof. Avishai Mandelbaum

2. 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. 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

4. 4 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. 5 Erlang-C (M/M/N) Background NμNμ (N-1)μ 01 N-1 N μ2μ2μNμNμ N+1 NμNμ Arrivals : Poisson(λ) Service: exp(μ) Number of Servers: N Utilization ρ ( =λ/Nμ ) <1 Steady State Erlang-C Formula

6. 6 Erlang-A (M/M/N+M) Background Nμ+θ (N-1)μ 01 N-1 N μ2μ2μNμNμ N+1 Nμ+2θ Arrivals : Poisson(λ) Service: exp(μ) Number of Servers: N Individual Patience: exp(θ) Erlang-A Formula Steady State always exists Offered Load per server ρ=λ/Nμ Note:

7. 7 ED QD QED Asymptotics: Background Operational Regimes ; 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 Define: = Offered Load. Erlang-C: Halfin-Whitt, 1981 Erlang-A: Garnett-Mandelbaum-Reiman, 2002

8. 8 Erlang- A/C: Excursions T = Avg. Busy Period T = Avg. Idle Period μ (N-1)μ 01 N-1 N μ2μ2μ NμNμ N+1 μ N Idle PeriodBusy Period N,N-1 N-1,N IdleBusy 00QED  0QD 0  ED lim rate

9. 9 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 λ Exponential service at rate μ Exponential Patience with rate θ ( Total = M/M/N(+M)) k d Preemptive Priority Non-Preemptive Priority High priority interrupts lower ones Service interruptions not allowed

10. 10 Some Notation: Priority Queues avg. waiting time of type k under Preemptive priority avg. waiting time of k first types under Preemptive priority avg. waiting time of all types under Preemptive priority avg. total number of delayed customers under Preemptive priority Similarly: Similarly: Non-Preemptive

11. 11 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 Similarly:

12. 12 Preemptive Priority Example: K=2 Calculation of average wait of class k,, k=1,2 Note: does not depend on service policy 1) 2)

13. 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! Step 1: Step 2: Step 3:

14. 14 Non-Preemptive Priority: Erlang-C Queues Kella & Yechielly (1985) proofs via model with vacations: Here- fraction of time spent with types 1, …, k Explanation

15. 15 Non-Preemptive Priority: Erlang-C Queues Avg. Queue length (given wait) M/M/N, Avg. Busy-Period duration M/M/1, By PASTA Erlang-C Diagram

16. 16 Non-Preemptive Priority: Erlang-A Queues The Highest Priority: (Delay probability does not depend on the service discipline)

17. 17 Nμ+3θ 1 1 Nμ+2θ Nμ+θ Nμ+2θ Nμ+θ Nμ+2θ Nμ+θ 2 2 2 2 2 2 1 0,0,0 1,0,02,0,0 N-1,0,0 N,0,0 N,1,0 N,2,0 N,3,0 N,1,1 N,0,1 N,0,2N,1,2 N,2,1N,3,1 N,2,2 N,3,2 1 1 1 2 1 1 1 2 μ2μ Nμ θ 2θ θθ Nμ+2θ Nμ+θ N,0 N,1N,2N,3 1 1 1 L L 1 + Non-Preemptive Priority: Transition-Rate Diagram

18. 18 Non-Preemptive Priority: K Types The Algorithm Step 1: Step 2: ”Merge” the first k types to a single type with Step 3:

19. 19 Towards : Example K=2 Non-Preemptive Preemptive

20. 20 Many Servers QED Example K=2 the same convergence rate! the same limit! QED Assume: Type 2 is not negligible: QD “QD | Wait”

21. 21 ED: the same convergence rate! (=1) the same limit! Many Servers ED Example K=2 Assume: Type 2 is not negligible:

22. 22 QED and ED with Abandonments: Summary of Results

23. 23 Erlang-C Erlang-A Many Servers, QED, ED Higher Priorities, Non-Preemptive: Erlang A = C that is Erlang-AErlang-C Higher priorities in Erlang –A enjoy QD regime (given they wait) hence “Erlang-C” performance

24. 24 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. 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