Overflow Networks: Approximations and Implications to Call-Center Outsourcing

Transcription

1 Overflow Networks: Approximations and Implications to Call-Center Outsourcing Itai Gurvich (Northwestern University) Joint work with Ohad Perry (CWI)

2 Call Centers with Overflow λ 1 λ 2 Source of complexity: dependence

3 Motivating Example 1 C O s (N O ) = capacity cost function for station O. W(t) = virtual waiting time at time t. (Similarly, W I (t) and W O (t)) W(t) = W I (t)1{x I (t) < N I + K I } + W O (t)1{x I (t) = N I + K I } min NO Cs O (N O ) s.t. E [f (W O (t))1{x I (t) = N I + K I }] α K λ θ θ λ 1 N O Z +, NI NO µ I µ O

4 Motivating Example 2 λ λ 1 λ 2 min i E[ T 0 C i(q π i (s))ds] s.t. π Π.

5 Motivating Example 2 λ λ 1 λ 2 min i E[ T 0 C i(q π i (s))ds] s.t. π Π. Optimal policy may benefit from state information on in-house

6 Related Literature Blocking and overflow: Exact characterization: Van Doorn ( 83); Approximations: Whitt ( 83), Koole et. al ( 00, 05); Heavy Traffic: Hunt and Kurtz ( 94), Koçaga and Ward ( 10), Pang et. al ( 07), Whitt ( 04); Outsourcing and optimization: Gans and Zhou ( 07); Chevalier et. al. ( 03, 04); Technical: Whitt ( 91), Bassamboo et. al ( 05), Perry and Whitt ( 10a); Glynn and Whitt ( 93), Perry and Whitt ( 10b);

7 Basic Model A(t) - number of arrivals to station I by time t: A(t) is a Poisson process with rate λ. A O (t) - number of overflowed calls by time t. K λ θ θ A I (t) = A(t) A O (t) - arrivals entering station I. X I (t), X O (t) - total number in respective system at t. NI NO K I - threshold in station I (K I 0). µ I µ O In isolation: station O is an GI/M/N + M queue.

8 Main Results A sequence of overflow networks in a many-server heavy-traffic regime 1 Functional Central Limit Theorem (FCLT) for Overflow Process 2 Pointwise Stationarity and Asymptotic Independence

9 Asymptotic (Heavy Traffic) Analysis We consider a sequence of networks indexed by arrival rate λ, with λ. Main Assumption: 1 Non-negligible overflow: K λ θ θ µ I NI λ + θki λ ν := lim λ λ < 1 NI NO µ I µ O 1 ν interpreted as rough estimate for the (steady-state) blocking probability

10 Asymptotic (Heavy Traffic) Analysis We consider a sequence of networks indexed by arrival rate λ, with λ. Main Assumption: 1 Non-negligible overflow: K λ θ θ µ I NI λ + θki λ ν := lim λ λ < 1 2 Sufficient capacity in station O: NI NO N λ O = λ µ IN λ I θ I K λ I µ O + o(λ) µ I µ O 1 ν interpreted as rough estimate for the (steady-state) blocking probability

11 First Result D λ I (t) = N λ I K λ I X λ I (t), D λ I (t) := Dλ I (t) λ, Â λ O(t) := Aλ O (t) (λ µ INI λ θki λ )t, X O(t) λ := Xλ O (t) λ µ INλ I θkλ I µ λ O λ

12 First Result D λ I (t) = N λ I K λ I X λ I (t), D λ I (t) := Dλ I (t) λ, Â λ O(t) := Aλ O (t) (λ µ INI λ θki λ )t, X O(t) λ := Xλ O (t) λ µ INλ I θkλ I µ λ O λ Theorem If D λ I (0) 0, then ( D λ I, Âλ O ) (0, σb), u.o.c., where B is a standard Brownian motion and σ 2 = 1 + ν.

13 First Result D λ I (t) = N λ I K λ I X λ I (t), D λ I (t) := Dλ I (t) λ, Â λ O(t) := Aλ O (t) (λ µ INI λ θki λ )t, X O(t) λ := Xλ O (t) λ µ INλ I θkλ I µ λ O λ Theorem If D λ I (0) 0, then ( D λ I, Âλ O ) (0, σb), u.o.c., where B is a standard Brownian motion and σ 2 = 1 + ν. Recall: ν := lim λ µ I N λ I + θk λ I λ

14 Outline of the proof: D λ I 0 D λ I (t) := Nλ I + KI λ XI λ(t) When close to threshold: rate µ I NI λ + θki λ, rate = λ. Slowing D λ I (t) down gives an M/M/1 with arrival rate = µ IN λ I + θk λ I λ ν < 1 and service rate = λ λ = 1 Let Q b (t) be M/M/1 with arrival rate ν and service rate 1. Then, {D λ I (t) : t [0, T)} {Q b(t) : t [0, λt)}. From extreme-value theory for M/M/1: sup t T D λ I (t) = O(log(λT))

15 Outline of the proof:  λ O σb A λ O(t) = t 0 1{D λ I (s) = 0}dA λ (s) D λ I completes O(λ) cycles over any time interval [s, t] Functional limits for the cumulative processes t 0 1{D λ I (s) = 0}ds (1 ν)t ( t ) λ 1{D λ I (s) = 0}ds (1 ν)t σb(t) 0 Functional limit for  λ O follows from that for t 0 1{Dλ I (s) = 0}ds

16 Outline of the proof:  λ O σb A λ O(t) = t 0 1{D λ I (s) = 0}dA λ (s) D λ I completes O(λ) cycles over any time interval [s, t] Functional limits for the cumulative processes t 0 1{D λ I (s) = 0}ds (1 ν)t ( t ) λ 1{D λ I (s) = 0}ds (1 ν)t σb(t) 0 Functional limit for  λ O follows from that for t 0 1{Dλ I (s) = 0}ds An Averaging Principle (AP):  λ O is driven by a process that moves at a different time scale

17 Implications Approximating (complicated) overflow process with a simple process: A λ O (t) (λ µ IN λ I θk λ I )t + λσb(t), with the approximation being asymptotically exact. A λ O (t) is close to a renewal process with mean inter-arrival time (λ µ I N λ I θ I K λ I ) 1 and squared coefficient of variation (SCV) λσ 2 λ µ I N λ I θk λ I σ2 (1 ν) 1.

18 Implications Approximating (complicated) overflow process with a simple process: A λ O (t) (λ µ IN λ I θk λ I )t + λσb(t), with the approximation being asymptotically exact. A λ O (t) is close to a renewal process with mean inter-arrival time (λ µ I N λ I θ I K λ I ) 1 and squared coefficient of variation (SCV) λσ 2 λ µ I N λ I θk λ I σ2 (1 ν) 1. Simpler than the original overflow renewal process.

19 Implications Cont. In isolation, station O is a GI/M/N + M queue. Using overflow convergence and known results for GI/M/N + M, ( D λ I, X O) λ (0, X O ) Asymptotic independence in a trivialized sense

20 Implications Cont. In isolation, station O is a GI/M/N + M queue. Using overflow convergence and known results for GI/M/N + M, ( D λ I, X O) λ (0, X O ) Asymptotic independence in a trivialized sense What does this limit imply for joint distributions?... not much... E[W λ (t)] = E[W λ I (t)1{x λ I (t) < N λ I + K λ I }] + E[W λ O(t)1{X λ I (t) = N λ I + K λ I }].

21 Independence of the Limits Independence of limits does not carry over to the pre-limits. Example: 1/ λ, w.p. 1/2 1, if Y λ > 0 Y λ := X λ := 0, w.p. 1/2 0, otherwise 1 w.p. 1/2 (Y λ, X λ ) (0, X), where X = 0 w.p. 1/2 Trivially, the limits 0 and X are independent. However, 1/2 = P{X λ > 0, Y λ > 0} P{X λ > 0}P{Y λ > 0} = 1/4, for all λ, no matter how large.

22 Asymptotic Independence Natural scale of station I = constant Natural scale of station O = λ Theorem (asymptotic independence) D λ I is asymptotically independent of X O λ, i.e, for all t > 0, { } { } } P D λ I (t) x, X O(t) λ y = P D λ I (t) x P { X O(t) λ y + o(1)

23 Asymptotic Independence Natural scale of station I = constant Natural scale of station O = λ Theorem (asymptotic independence) D λ I is asymptotically independent of X O λ, i.e, for all t > 0, { } { } } P D λ I (t) x, X O(t) λ y = P D λ I (t) x P { X O(t) λ y + o(1) Note that X λ O is scaled, but Dλ I is not (requires refined analysis).

24 Main Idea of the Proof Showing asymptotic independence of the sequence in n via asymptotic independence of the process in t

25 Main Idea of the Proof Showing asymptotic independence of the sequence in n via asymptotic independence of the process in t (*) Recall that {D λ I (s) : t s t + ɛ} {Q b(s) : t s t + λɛ} for λ large, with Q b denoting a M/M/1.

26 Main Idea of the Proof Showing asymptotic independence of the sequence in n via asymptotic independence of the process in t (*) Recall that {D λ I (s) : t s t + ɛ} {Q b(s) : t s t + λɛ} for λ large, with Q b denoting a M/M/1. (**) Q b (t + λɛ) Q b ( ) as λ for all ɛ > 0.

27 Main Idea of the Proof Showing asymptotic independence of the sequence in n via asymptotic independence of the process in t (*) Recall that {D λ I (s) : t s t + ɛ} {Q b(s) : t s t + λɛ} for λ large, with Q b denoting a M/M/1. (**) Q b (t + λɛ) Q b ( ) as λ for all ɛ > 0. Steady state Q b ( ) is independent of Q b (t).

28 Main Idea of the Proof Showing asymptotic independence of the sequence in n via asymptotic independence of the process in t (*) Recall that {D λ I (s) : t s t + ɛ} {Q b(s) : t s t + λɛ} for λ large, with Q b denoting a M/M/1. (**) Q b (t + λɛ) Q b ( ) as λ for all ɛ > 0. Steady state Q b ( ) is independent of Q b (t). X λ O has a continuous limit hence hardly changes within ɛ, X λ O(t + ɛ) X λ O(t).

29 Pointwise Stationarity The following pointwise stationarity follows from (*) and (**): Theorem (pointwise stationarity) D λ I (t) Q b( ) in R as λ.

30 Pointwise Stationarity The following pointwise stationarity follows from (*) and (**): Theorem (pointwise stationarity) D λ I (t) Q b( ) in R as λ. Pointwise stationarity and asymptotic independence allow us to obtain performance metrics by treating Station I as a stationary M/M/N/K + M queue Station O as a GI/M/N + M queue (that is independent of station I) P{D λ I (t) d, XO(t) λ q} = P{D λ I ( ) d}p{xo(t) λ q} Overflow approximation simplifies analysis of station O

31 Waiting Times and Asymptotic ASTA w λ k, wλ I,k, wλ O,k - waiting time of kth arrival to respective station. f is a continuous and bounded function or of the form f (x) := 1{x > τ}. Theorem (asymptotic finite-horizon ASTA) For all t > 0, lim E 1 λ A λ (t) A λ (t) k=1 f (w λ k ) = ν 1 t t 0 [ ] E f (Ŵ I (s)) ds+(1 ν) 1 t t 0 [ ] E f (Ŵ O (s))ds. where Ŵ O is the diffusion limit of the virtual waiting-time process in the GI/M/N + M queue and Ŵ I K I.

32 Generalizing to multiple classes λ λ 1 λ 2 min i E[ T 0 C i(q π i (s))ds] s.t. π Π. Theorem: Benefit from in-house state information is marginal.

33 Summary Motivated by an outsourcing problem, we considered an overflow system: from M/M/N I /K I + M to G/M/N O + M. Under a resource pooling condition our heavy traffic analysis: provides a simple approximation for the overflow renewal process, which is asymptotically correct. proves that in-house is asymptotically independent of outsourcer. Proofs build on a separation of time scales and a resulting AP and pointwise stationarity. Results are applied to waiting times and virtual waiting times. Generalized to more complicated systems (if queues are C-tight).

34 Questions?