Making Delay Announcements

Transcription

1 Making Delay Announcements Performance Impact and Predicting Queueing Delays Ward Whitt With Mor Armony, Rouba Ibrahim and Nahum Shimkin March 7, 2012

2 Last Class 1 The Overloaded G/GI/s + GI Fluid Queue Model (W 2, Operations Research, 2006) 2 The G t /GI/s t + GI Fluid Model with Alternating Overloaded and Underloaded Intervals (Yunan Liu & W 2, Queueing Systems, 2012)

3 Today: Application of Fluid Models to Delay Prediction 1 The Performance Impact of Delay Announcements to New Arrivals (Mor Armony, Nahum Shimkin & W 2, Operations Research, 2009) 2 Real-Time Delay Predictors (including Time-Varying Arrivals) (Rouba Ibrahim & W 2, Operations Research, 2011)

4 1. Review: The G/GI/s+GI Fluid Model Approximation for the G/GI/s + GI Stochastic Queueing Model service facility λ arrival process waiting room departures abandonment F(x) = P(T a x) (T a time to abandon) s servers G(x) = P(T s x) (T s service time)

5 Many-Server Heavy-Traffic (MSHT) Limit Increasing Scale Increasing Scale a sequence of G/GI/s + GI models indexed by n, arrival rate grows: λ n /n λ as n, number of servers grows: s n /n s as n, service-time cdf G and patience cdf F held fixed independent of n with mean service time 1: µ 1 0 x dg(x) 1.

6 The G/GI/s+GI Fluid Model Model data: (λ, s, G, F) and initial conditions. service facility λ input flow waiting room departure flow abandonment F(x) = proportion abandoning by time x capacity s G(x) = proportion served by time x

7 The Overloaded Fluid Model in Steady State fluid density arriving time t in the past in service sg c (u) in queue F c (t) s w + u w time t 0

8 2. The Performance Impact of Delay Announcements make delay announcement (single-number w) to new arrivals helpful with invisible queues (e.g., waiting on the telephone) But we need to consider the customer response Assume: balk (leave immediately) with probability B(w) If join, abandon before time t with probability F(t w) What to announce? Delay of Last to Enter Service (LES) What is the performance impact of the announcement?

9 Use Fluid Model to Study Performance Impact Direct Response to Delay Announcement in service in queue s w~ ) ( b sg c (u) w~ b u w ~ b w ~ a 0

10 Equilibrium Delay in service sg c (u) in queue F c (t) ( w~ ) s w ~ e ) ( 1 w~ u ~w w ~ e ~w 2 time t 0 1 1

11 Use Fluid Model to Study Convergence Cycling Around the Equilibrium Delay in service in queue s ( w 1 ) ( w~ e ) ( w 2 ) s w ~ w ~ w time t 2 e w 1 0 n

12 Conclusions of Study, Armony et al Under general conditions, there exists a unique equilibrium delay in the fluid model. Direct iteration w n+1 = d(w n ) as shown above can lead to oscillations. Damped iterations produce convergence: w n+1 = pd(w n ) + (1 p)w n LES delay and fluid equilibrium delay both work well in simulations But LES more robust, does not depend on the model.

13 3. Alternative Real-Time Delay Predictors Now considered without customer response Delay Experienced by the Last to Enter Service (LES) Elapsed Delay of the Customer at the Head of Line (HOL) Standard Simple Queue-Length-Based Delay Predictor QL s Use the Fluid Model to Develop Refined Predictors

14 The Standard Simple Queue-Length-Based Delay Predictor Servers Queue New arrival n customers in line upon arrival s = number of agents, and µ 1 = mean service time θ QLs (n) (n + 1) µ 1 s

15 The Head-of-Line (HOL) Delay Predictor Servers Queue New arrival at t HOL Elapsed delay at t is equal to w w = elapsed delay of HOL customer (similar to LES delay) θ HOL (w) w

16 Actual Random Delays After Prediction W HOL (w): distributed as the delay of a new arriving customer given that: (i) there is a customer at the HOL upon arrival (non-restrictive) (ii) elapsed delay of HOL customer is equal to w W Q (n): distributed as the delay of a new arriving customer given that: (i) the customer has to wait (ii) the customer finds n customers in line upon arrival We announce θ HOL (w) w w is a single-number prediction of the random variable W HOL (w).

17 Quantifying The Accuracy of the Predictors Mean Squared Error (MSE) MSE(θ QLs (n)) = E[(W Q (n) θ QLs (n)) 2 ] E[MSE(θ QLs (Q w ))] = MSE(θ QLs (n))p[q w = n] n=0 Q w has the conditional distribution of the steady-state QL upon arrival given that the customer must wait.

18 How to Evaluate Predictors with Simulation Simulation Estimate of MSE: Average Squared Error (ASE) ASE 1 k k (p i d i ) 2 (k = sample size) i=1 p i = predicted delay for customer i (p i > 0) d i = actual delay (or potential delay with abandonments) Root Relative Average Squared Error (RASE) RASE ASE 1 k k i=1 d i

19 QL s in the GI/M/s Model n+1 W Q (n) = i=1 where V i i.i.d. exponential with mean (sµ) 1 n+1 E[W Q (n)] = E[V i ] = i=1 n+1 i=1 V i 1 sµ = n + 1 sµ θ QL s (n) n+1 MSE(θ QLs (n)) = Var[W Q (n)] = Var[V i ] = i=1 n+1 i=1 θ QLs (n) minimizes the MSE! 1 s 2 µ 2 = n + 1 s 2 µ 2

20 HOL in the M/M/s Model W HOL (w) = A(w)+2 i=1 V i where V i i.i.d. exponential with mean (sµ) 1 E[W HOL (w)] = E A(w)+2 V i i=1 = E[A(w) + 2]E[V] w θ HOL (w) Var[W HOL (w)] = E[A(w) + 2Var[V] + Var[A(w)](E[V]) 2

21 Simulations for the GI/M/s Model: Poisson Arrivals In Tables: ASE s in units of 10 3 (RASE in %); ρ = λ/sµ; c 2 a = Var/mean 2. M/M/100 ρ QL s HOL HOL/QL s (c 2 a + 1)/ρ (14%) 10.2 (20%) (22%) 4.27 (32%) (26%) 3.08 (39%) (32%) 2.19 (47%) Similar for other renewal arrival processes: ratio (c 2 a + 1)/ρ.

22 Simulations for the GI/M/s Model: Deterministic Arrivals In Tables: ASE s in units of 10 3 (RASE in %); ρ = λ/sµ; c 2 a = Var/mean 2. D/M/100 ρ QL s HOL HOL/QL s (c 2 a + 1)/ρ (20%) 2.62 (21%) (32%) 1.15 (34%) (37%) (41%) (44%) (50%)

23 Abandonments: Simulations for the M/M/s + M Model HOL QL s s ASE in the M/M/s+M Model with ρ = s ASE s

24 W Q (n) in the GI/M/s + M Model W Q (n): distributed as the potential delay of a new arriving customer given that: (i) the customer has to wait (ii) the customer finds n customers in line upon arrival W Q (n) = n i=0 X i where X i independent exponential with mean (sµ + iν) 1

25 Markovian QL Predictor (QL m ) The Markovian Queue-Length Predictor (QL m ) θ QLm (n) = n i=0 1 sµ + iν QL m in the GI/M/s + M Model θ QLm (n) = n 1/(sµ + iν) = E[W Q (n)] i=0 θ QLm (n) minimizes the MSE!

26 Refined QL Predictor for Same M/M/s + M Model QL m HOL QL s s ASE in the M/M/s+M Model with ρ = s ASE s

27 But when the Abandonment Distribution is Not Exponential 7 6 HOL QL m s ASE in the M/M/s+E 10 Model with ρ = s ASE s

28 s ASE QL a QL r QL m HOL QL s s ASE in the M/M/s+H 2 Model with ρ = s

29 7 s ASE in the M/E 10 /s+e 10 Model with ρ = QL a QL r HOL QL m s ASE s

30 Time-Varying Arrival Rates arrivals per hour to a medium-sized financial-services call center

31 HOL Delay Prediction in the M t /M/100 Model Potential Delays and HOL Predictions in the M(t)/M/100 Model with α = 0.5 and E[S] = 5 minutes Delays HOL Delays time Arrival process: Nonhomogeneous Poisson with rate λ(t) sinusoidal λ(t) = λ + α λ sin(γt), ρ = λ/sµ = 0.95

32 HOL Delay Prediction With Constant Arrival Rate Actual Delays and HOL Estimates in the M/M/100 Model with ρ = 0.95 Delays HOL Delays time Arrival process: homogeneous Poisson with rate λ ρ = λ/sµ = 0.95

33 Problem: Time Lag in HOL Delay HOL delay was potential delay of arrival in the past. Use fluid model to create refined predictor. v(t) potential delay of new arrival in fluid model at time t w(t) HOL delay in the fluid model at time t θ HOLr (w, t) = v(t) w(t) w, where w is observed HOL delay at time t in actual system.

34 HOL r Delay Prediction in the M t /M/100 Model Actual Delays, HOL, and HOL r in the M t /M/100 Model with α = 0.5 and E[S] = 5 minutes Delays HOL r HOL 25 Delays time Arrival process: Nonhomogeneous Poisson with rate λ(t) λ(t) = λ + α λ sin(γt), ρ = λ/sµ = 0.95

35 SUMMARY 1 We have discussed the fluid approximation for many-server queues. 2 It can be used to study the performance impact of delay announcements. 3 It can be used to help make better delay predictions.

36 The End

37 References

38 Other Papers with Rouba Ibrahim Real-Time Delay Estimation Based on Delay History. Manuf. Serv. Opns. Mgt. 11 (2009) Real-Time Delay Estimation in Overloaded Multiserver Queues with Abandonment. Man. Sci. 10 (2009) Real-Time Delay Estimation Based on Delay History in Many-Server Queues with Time-Varying Arrivals. Prod. Opns. Mgt..