Modeling Service Networks with Time-Varying Demand
|
|
- Lambert Bates
- 6 years ago
- Views:
Transcription
1 Modeling Service with Time-Varying Demand - Performance Approximations and Staffing Controls (with Beixiang He, Liam Huang, Korhan Aras, Ward Whitt) Department of Industrial and Systems Engineering NC State University A survey on recent results of time-varying queueing models
2 Education B.E. in Electrical Engineering, Tsinghua Univ., Beijing, 2002 M.S. & Ph.D. in Oper. Res., Columbia Univ., NY, 2007, 2011 Research Interests Method: stochastic modeling, applied prob., queueing theory Application: call centers, health care, manufacturing systems When I m not teaching or writing papers...
3 Current Ph.D. students at NCSU Beixiang He (Graduating in August 2014) Korhan Aras (Graduating in Spring 2015) Liam Huang, Yao Yu (Ongoing) Former Ph.D. advisor Prof. Ward Whitt (Columbia University)
4 Nonstationary Systems: Time-Varying Arrivals service call center emergency room Green et al. (2007) Yom-Tov and Mandelbaum (2011)
5 Realistic Models Features: General Distributions Non-exponential service and abandonment service service and abandonment abandonment Brown et al. (2005)
6 The Base Queueing Model G t /GI/s t + GI Time-varying arrival rate λ(t) (the G t ) (e.g., non-homogeneous Poisson, M t ) I.I.D. service times G(x) P(S x) (the first GI) Time-varying staffing level s(t) (the s t ) I.I.D. abandonment times F (x) P(A x) (the +GI) First-Come First-Served (FCFS) Unlimited waiting capacity
7 The Base Queueing Model G t /GI/s t + GI Time-varying arrival rate λ(t) (the G t ) (e.g., non-homogeneous Poisson, M t ) I.I.D. service times G(x) P(S x) (the first GI) Time-varying staffing level s(t) (the s t ) I.I.D. abandonment times F (x) P(A x) (the +GI) First-Come First-Served (FCFS) Unlimited waiting capacity Textbook models: M/M/1 No longer useful!!
8 The Base Queueing Model Performance measures Q(t) and B(t): number waiting in queue and in service at t X (t) Q(t) + B(t): total number in system W (t) and V (t): head-of-line and potential waiting time at t Exact analysis HARD! Time-varying arrival rate and staffing function Customer abandonment Large scale (for large λ(t) and s(t)) Non-Markovian probability structure: non-poisson, non-exponential Complicated network structure
9 Many-Server Heavy-Traffic Approximations A sequence of queues indexed by n arrival rate: λ n (t) = n λ(t) number of servers: s n (t) = n s(t) system size grows with n customer individual behavior unscaled: service cdf G and patience cdf F held fixed independent of n
10 Many-Server Heavy-Traffic Limits Fluid Limit LLN scaling: Qn (t) Qn(t) n, B n (t) Bn(t) n, X n (t) Xn(t) n FSLLN: ( Qn, B n, X n, W n ) (Q, B, X, W ) in D 4, as n
11 Many-Server Heavy-Traffic Limits Fluid Limit LLN scaling: Qn (t) Qn(t) n, B n (t) Bn(t) n, X n (t) Xn(t) n FSLLN: ( Qn, B n, X n, W n ) (Q, B, X, W ) in D 4, as n Diffusion Limit CLT scaling: ˆQ n (t) n ( Q n (t) Q(t) ) Qn(t) n Q(t) = n, Ŵ n (t) n (W n (t) W (t)) FCLT: ( ˆQn, ˆB n, ˆX Ŵn) ( ) n, ˆQ, ˆB, ˆX, Ŵ in D 4, as n
12 Many-Server Heavy-Traffic Limits Fluid Limit LLN scaling: Qn (t) Qn(t) n, B n (t) Bn(t) n, X n (t) Xn(t) n FSLLN: ( Qn, B n, X n, W n ) (Q, B, X, W ) in D 4, as n Diffusion Limit CLT scaling: ˆQ n (t) n ( Q n (t) Q(t) ) Qn(t) n Q(t) = n, Ŵ n (t) n (W n (t) W (t)) FCLT: ( ˆQn, ˆB n, ˆX Ŵn) ( ) n, ˆQ, ˆB, ˆX, Ŵ in D 4, as n Approximations Q n (t) = n Q(t) + n ˆQ(t) + o( ) n) d N (n Q(t), n σ 2ˆQ (t) ( ) W n (t) = W (t) + Ŵ (t) n + o( 1 n ) d N W (t), 1n σ2ŵ (t)
13 MSHT Fluid Limits Fluid Limit LLN scaling: Qn (t) Qn(t) n, B n (t) Bn(t) n, X n (t) Xn(t) n FSLLN: ( Qn, B n, X n, W n ) (Q, B, X, W ) in D 4, as n Diffusion Limit CLT scaling: ˆQ n (t) n ( Q n (t) Q(t) ) Qn(t) n Q(t) = n, Ŵ n (t) n (W n (t) W (t)) FCLT: ( ˆQn, ˆB n, ˆX Ŵn) ( ) n, ˆQ, ˆB, ˆX, Ŵ in D 4, as n Approximations Q n (t) = n Q(t) + n ˆQ(t) + o( ) n) d N (n Q(t), n σ 2ˆQ (t) ( ) W n (t) = W (t) + Ŵ (t) n + o( 1 n ) d N W (t), 1n σ2ŵ (t)
14 What Are s?
15 MSHT Fluid Limit (FWLLN) random deterministic; discrete continuous; finite infinitely divisible.
16 Realistic Models Features: Network Structure Markovian routing Once finishing at one station, flip a coin to decide next station Pros: capture randomness in routing; tractable due to Markov property Cons: completely ignore service history; too crude to approximate real systems Prescribed paths Each customer follows a prescribed itineraries/paths Customers with same itineraries belong to same class Pros: characterizes different classes, capture service history Cons: difficult to analyze (no Markov property)
17
18 with
19
20 : A Dynamical System Key functions: fluid densities Q(t, y) : quantity of fluid in queue for up to y at t Q(t) : y q(t, x)dx 0 B(t, y) : quantity of fluid in service for up to y at t B(t) : y b(t, x)dx 0
21 : A Dynamical System Key functions: fluid densities Q(t, y) : quantity of fluid in queue for up to y at t Q(t) : y q(t, x)dx 0 B(t, y) : quantity of fluid in service for up to y at t B(t) : y b(t, x)dx 0 Fluid densities: transport PDE q(t,x) t + q(t,x) b(t,x) t x = h F (x) q(t, x), 0 x w(t), t 0. + b(t,x) t = h G (x) b(t, x), w(t)x 0, t 0. Fluid waiting time: ODE head-of-line (HOL) and potential waiting times at t w (t) = 1 b(t,0) q(t,w(t)) and v (t) = q(t+v(t),v(t)) b(t+v(t),0) 1 Rate into service: fixed-point equation (FPE) b(t, 0) = S (t) + t b(t x, 0)g(x)dx, 0 x t. 0
22 Algorithm for the (G t /GI/s t + GI) m /M t Fluid Network
23 A Non-Markovian Example M t /H 2 /s t + E 2 fluid model λ(t) = sin(t) S = 1 (note: not a single-server queue) H 2 service: p = 0.11, µ 1 = 0.23, µ 2 = 1.77 (Cs 2 = 4) E 2 abandonment: A = X 1 + X 2, where X i i.i.d. exp(1) System initially empty λ(t) and S will be scaled by n!
24 Fluid Algorithm: Alternating between OL and UL
25 Fluid Algorithm: Alternating between OL and UL
26 Fluid Algorithm: Alternating between OL and UL
27 Fluid Algorithm: Alternating between OL and UL
28 Fluid Algorithm: Alternating between OL and UL
29 Fluid Algorithm: Alternating between OL and UL
30 Simulation Comparisons M t /H 2 /s t + E 2 queueing model n = 20, 100, 2000 λ n (t) = n λ(t) = n n sin(t) S n (t) = n S(t) = n Want to see When n is large: ( Qn(t) n, Bn(t) n When n is small: ( E[Qn(t)] n, E[Bn(t)] n, Xn(t) n, W n (t), E[Xn(t)] n, E[W n (t)] ) (Q(t), B(t), X (t), w(t)) ) (Q(t), B(t), X (t), w(t))
31 Simulation Comparisons: M t /H 2 /s t + E 2 n = 100 and 3 sample paths
32 Simulation Comparisons: M t /H 2 /s t + E 2 n = 2000 and a single sample path
33 Simulation Comparisons: M t /H 2 /s t + E 2 n = 100 and a average of 100 sample paths
34 Example: A Network of Two Queues (M t /LN/s t + E 2 ) 2 /M t fluid network Sinusoidal arrival: λ (0) 1 (t) = n( sin(t)), Sinusoidal arrival: λ (0) 2 (t) = n( sin(t 3)) Constant staffing: s 1 (t) = n, s 2 (t) = 2n Lognormal service: Ḡ 1 LN(1, 2), Ḡ2 LN(2, 8) Erlang abandonment: F 1 E 2 (0.5), F 2 E 2 (0.3) [ ] Routing probability: P(t) System initially empty
35 Example: (M t /LN/s t + E 2 ) 2 /M t Simulation comparison: n = 2000, one path
36 Example: (M t /LN/s t + E 2 ) 2 /M t Simulation comparison: n = 50, 2000 paths
37 Example: (M t /LN/s t + E 2 ) 2 /M t Algorithm convergence ɛ: error tolerance (10 9 to 0.1) I(ɛ): # of iterations T (ɛ): computation time (seconds) in MatLab
38 Example: An (M t /M/s t + M) 10 /M t Network For 1 i, j 10, Sinusoidal arrival: λ (0) i (t) = a i + b i sin(c i t + φ i ), a i = 0.5, b i = i 10 a i, φ i = π ( 1.5 i Constant staffing: s i (t) = 1 Exponential service: Ḡ i (x) = e µ i x, µ i = 1 10), Exponential abandonment: F i (x) = e θ i x, θ i = 0.5 Routing probability: P i,j (t) 1 20 System initially empty
39 Example: Fluid Paths of (M t /M/s t + M) 10 /M t
40 Large Scale Network: Running Time Complexity
41 MSHT (FCLT) Fluid Limit LLN scaling: Qn (t) Qn(t) n, B n (t) Bn(t) n, X n (t) Xn(t) n FSLLN: ( Qn, B n, X n, W n ) (Q, B, X, W ) in D 4, as n Diffusion Limit CLT scaling: ˆQ n (t) n ( Q n (t) Q(t) ) Qn(t) n Q(t) = n, Ŵ n (t) n (W n (t) W (t)) FCLT: ( ˆQn, ˆB n, ˆX Ŵn) ( ) n, ˆQ, ˆB, ˆX, Ŵ in D 4, as n Approximations Q n (t) = n Q(t) + n ˆQ(t) + o( ) n) d N (n Q(t), n σ 2ˆQ (t) ) W n (t) = W (t) + Ŵ (t) n + o( 1 n ) d N (W (t), 1n σ2ŵ (t)
42
43 Characterizing the Diffusion Process Partition {1,..., m} = O U O {1, 2,..., m }, U {m + 1, m + 2,..., m}. For OL queues 1,..., m A Stochastic Differential Equation (SDE): dŵ(t) = H(t)Ŵ(t)dt + J s(t)db s(t) + J a(t)db a(t) + J λ (t)db λ (t) dŵ (t) = H(t)Ŵ(t)dt + J(t)dB (t) Bλ : m-dim BM (arrival process) Ba: m -dim BM (abandonment times) Bs: (m + 1) 2 -dim BM (service times) H, Js, J a, J λ and J : analytic matrices of λ, s, F, µ, Cλ, 2 P and fluid functions Var(Ŵ(t)) = t 0 Ĵs(u)ĴT s (u) + Ĵa(u)ĴT a (u) + Ĵλ(u)ĴT λ (u) du ˆQ(t): Brownian integrals w.r.t. B λ, B a and B s For UL queues m + 1,..., m d ˆB(t) = H (t)ˆb(t)dt + J s (t)db s (t) + J λ (t)db λ(t)
44 Special Case: One-Queue Model dŵ (t) = H(t)Ŵ (t)dt + J s(t)db s (t) + J a (t)db a (t) + J λ (t)db λ (t) dŵ (t) = H(t)Ŵ (t)dt + J (t)db (t) σ 2 (t) Var(Ŵ (t)) = ) t (Ĵ2 Ŵ 0 s (t, u) + Ĵa 2 (t, u) + Ĵλ 2(t, u) du ( ) H(t) = (1 w (t)) λ (t w(t)) λ(t w(t)) + h F (w(t)) J s (t) = J a (t) = b(t,0) s (t) λ(t w(t)) F (w(t)) F (w(t))b(t,0) λ(t w(t)) F (w(t)) J λ (t) = C λ J (t) = Want to see ) Var (Ŵn (t) ( ) Var ˆQn (t) F (w(t))b(t,0) λ(t w(t)) F (w(t)) b(t,0) s (t)+(f (w(t))+cλ 2 F (w(t))) b(t,0) λ(t w(t)) F (w(t)) ( ) σ 2 (t), Var ˆVn (t) σ 2ˆV (t), Ŵ σ (t) 2ˆQ
45 Example: M t /M/s t + H 2 in Both UL and OL Intervals λ(t) = n( sin(t)), s(t) = n, µ = 1, θ = 0.5 n = 2000 and 500 sample path
46 Engineering Refinement for Smaller n λ(t) = n( sin(t)), s(t) = n, µ = 1, θ = 0.5 n = 100 and 2000 sample path
47 Engineering Refinement for Smaller n λ(t) = n( sin(t)), s(t) = n, µ = 1, θ = 0.5 n = 20 and 5000 sample path
48
49 Prescribed Routing Paths
50 Multiclass Queueing Network with (G t /GI + GI) n /s m t /PRP n customer classes and m service pools Class i (1 i n) characterized by a deterministic path πi with π i (k) denotes the destination at step k (the PRP) an external arrival rate λi (the G t) non-exponential service dist. Gi,k (the first GI) non-exponential abandonment dist. Fi,k (the +GI) Time-varying staffing level s j (t) at queue j, 1 j m (the s t ) First-Come First-Served (FCFS) at each queue j Unlimited waiting capacity
51 Multiclass Queueing Network with A two-class two-queue example Class 1: π 1 = [1, 2, 1] Class 2: π 2 = [2].
52 MSHT Fluid Limits Fluid Limit LLN scaling: Qn (t) Qn(t) n, B n (t) Bn(t) n, X n (t) Xn(t) n FSLLN: ( Qn, B n, X n, W n ) (Q, B, X, W ) in D 4, as n Diffusion Limit CLT scaling: ˆQ n (t) n ( Q n (t) Q(t) ) Qn(t) n Q(t) = n, Ŵ n (t) n (W n (t) W (t)) FCLT: ( ˆQn, ˆB n, ˆX Ŵn) ( ) n, ˆQ, ˆB, ˆX, Ŵ in D 4, as n Approximations Q n (t) = n Q(t) + n ˆQ(t) + o( ) n) d N (n Q(t), n σ 2ˆQ (t) ( ) W n (t) = W (t) + Ŵ (t) n + o( 1 n ) d N W (t), 1n σ2ŵ (t)
53 Algorithm for the (G t /GI + GI) n /s m t /PRP Fluid Network
54 Multiclass Queueing Network with A two-class two-queue example
55 Example: An (M t /LN + E 2 ) 2 /s 2 t /PRP Network Simulation comparison
56 Achieving Time-Stable Performance Shi et al. (2014) Avoid large delay spikes Smooth performance level across time System becomes stationary
57 Staffing to Achieve Time-Stable Performance Design staffing and shifts Meet service level agreements P(waiting < 2 mins)>0.8 E(wait) 3 mins = 0.05 hr P(Abandonment)<0.02
58 Mean Delay At each station j, design staffing s j (t) to stabilize delay at w j Approximate other performance functions (e.g., queue length) Treating both high QoS (small w j ) and low QoS (big w j ) Class dependent service levels calls: to reply in minutes (small wj ) s: to reply in hours (big wj )
59 Delayed Infinite-Server (DIS) Approx. for M t /GI /s t + GI DIS staffing: mean # of busy servers s(t) = E[B(t)] F (w) (t w) + 0 λ(t w x)ḡ(x)dx Approximations for other performance measures t E[Q(t)] = λ(t x) F (x)dx (t w) + P(A < W (t)) = α F (w) Asymptotic stability: when scale becomes large sup 0<t T E[W (t)] w 0, sup 0<t T E[P(A < W (t)) α] 0
60 DIS Approx. for (G t /GI + GI) n /s m t /PRP DIS staffing formulas construct DIS approx. for each class i, step k m i,k (t) F (t wj ) + (w j ) λ i,k (t w j x)ḡi,k(x)dx 0 staffing at queue j: sum of all the required staffing for each type of customers that visits queue j s j (t) = N m i m i,k (t) 1 {πi (k)=j} i=1 k=1 Approximations for other performance measures t E[Qi,k (t)] λ i,k (t x) F i,k (x)dx (t w j ) + N m i E[Qj (t)] E[Q i,k (t)] 1 {πi (k)=j} αi,k (t) i=1 k=1 n F i,k (w j ) 1 {πi (k)=j} j=1
61 DIS Approx. for (G t /GI + GI) n /s m t /PRP Asymptotic stability: when scale becomes large sup E[W j (t)] w j 0 0<t T sup Pi,k ab (t) α j] 0 0<t T
62 A Test Example: (M t /GI + GI) 2 /s 2 t /PRP Class i 1 2 Arrival Rate λ (0) i (t) sin t cos(t + 0.5) Path π i π 1 = [1, 2, 1], N 1 = 3 π 2 = [2], N 2 = 1 Stage k, 1 k N i Service distribution LN(0.5, 0.75) c1,1 s =1.5 LN(1.5, 3) 1,2 =2 LN(1, 3) 1,3 =3 LN(2, 5) 2,1 =2.5 Abandonment distribution H 2 (0.15, 0.3) H 2 (1.15, 2.5) H 2 (1, 3) H 2 (0.25, 0.6) c1,1 a =2 ca 1,2 =2.17 ca 1,3 =3 ca 2,1 =2.4 LN(µ, σ 2 ) g(x) = 1 x (ln x µ) 2 2πσ e 2σ 2 H 2 (µ, σ 2 ) f (x) = pλ 1 e λ1x + (1 p)λ 2 e λ2x where p = µ2 +σ 2 + σ 4 µ 4 2(σ 2 +µ 2 ), λ 1 = 2p/µ, λ 2 = 2(1 p)/µ
63 A Test Example: (M t /GI + GI) 2 /s 2 t /PRP Arrival rates
64 A Test Example: (M t /GI + GI) 2 /s 2 t /PRP Low QoS: w 1 = 0.1, w 2 = 0.3
65 A Test Example: (M t /GI + GI) 2 /s 2 t /PRP Low QoS: w 1 = 0.1, w 2 = 0.3
66 A Test Example: (M t /GI + GI) 2 /s 2 t /PRP Low QoS: w 1 = 0.1, w 2 = 0.3
67 A Test Example: (M t /GI + GI) 2 /s 2 t /PRP High QoS: w 1 = 0.005, w 2 = 0.02
68 A Test Example: (M t /GI + GI) 2 /s 2 t /PRP High QoS: w 1 = 0.005, w 2 = 0.02
69 A Test Example: (M t /GI + GI) 2 /s 2 t /PRP Mixed: w 1 = (high QoS), w 2 = 0.1 (low QoS)
70 A Test Example: (M t /GI + GI) 2 /s 2 t /PRP Mixed: w 1 = (high QoS), w 2 = 0.1 (low QoS)
71 Summary of Fluid Summary Markovian routing: (G t /GI/s t + GI) m /M t network Deterministic paths: (G t /GI + GI) m /s n t /PRP network Fluid (FWLLN) and diffusion (FCLT) approximations Staffing to achieve time-stable performance Simulations comparisons to verify effectiveness Future work Fit into realistic applications (e.g., health care) Incorporate both uncertainty and history
72 References [1] He & Liu, the Tail Probability of Delay in Service Systems with Time-Varying Demand. Submitted to Operations Research (2014) [2] Aras, Liu & Whitt, Heavy-Traffic Limit for the Initial Content Process. Submitted to Mathematics of Operations Research (2014) [3] Liu & Whitt, Stabilizing Performance in Many-Server Queues with Time-Varying Arrivals and Customer Feedback. Submitted to Operations Research (2014) [4] Liu & Whitt, Many-Server Heavy-Traffic Limits for Queues with Time-Varying Parameters. Annals of Applied Probability 24(1), (2014) [5] Liu & Whitt, Algorithms for Time-Varying of Many-Server Fluid Queues. INFORMS Journal on Computing 26(1), (2014) [6] Liu & Whitt, The G t/gi /s t + GI Many-Server Fluid Queue. Queueing Systems 71(4), (2012) [7] Liu & Whitt, Stabilizing Customer Abandonment in Many-Server Queues with Time-Varying Arrivals. Operations Research 60(6) (2012) [8] Liu & Whitt, A Network of Time-Varying Many-Server Fluid Queues with Customer Abandonment. Operations Research 59(4) (2011) Available at
73 THANK YOU!
74 The G t /GI /s t + GI Queueing Model Constructing the G t arrivals Deterministic variability: time-varying λ(t) (avg. behavior) Stochastic variability: constant c λ (magnitude of variance) Composition methods Step 1: generate rate-1 (equilibrium) renewal process N 0 (t), with variance of interrenewal time Var(X ) = cλ 2 Step 2: set N(t) N 0 (Λ(t)), with Λ(t) t 0 λ(u)du Properties: Mean value: E[N(t)] = Λ(t), Variance-to-mean ratio: Var(N(t))/E[N(t)] = Var(N(t))/Λ(t) c λ (Liu et al. 2014)
Making Delay Announcements
Making Delay Announcements Performance Impact and Predicting Queueing Delays Ward Whitt With Mor Armony, Rouba Ibrahim and Nahum Shimkin March 7, 2012 Last Class 1 The Overloaded G/GI/s + GI Fluid Queue
More informationMANY-SERVER HEAVY-TRAFFIC LIMIT FOR QUEUES WITH TIME-VARYING PARAMETERS 1
The Annals of Applied Probability 214, Vol. 24, No. 1, 378 421 DOI: 1.1214/13-AAP927 Institute of Mathematical Statistics, 214 MANY-SERVER HEAVY-TRAFFIC LIMIT FOR QUEUES WITH TIME-VARYING PARAMETERS 1
More informationStabilizing Customer Abandonment in Many-Server Queues with Time-Varying Arrivals
OPERATIONS RESEARCH Vol. 6, No. 6, November December 212, pp. 1551 1564 ISSN 3-364X (print) ISSN 1526-5463 (online) http://dx.doi.org/1.1287/opre.112.114 212 INFORMS Stabilizing Customer Abandonment in
More informationThe G t /GI/s t + GI Many-Server Fluid Queue: Longer Online Version with Appendix
Queueing Systems manuscript No. (will be inserted by the editor) January 4, 22 The G t /GI/s t + GI Many-Server Fluid Queue: Longer Online Version with Appendix Yunan Liu Ward Whitt Received: date / Accepted:
More informationOnline Supplement to Delay-Based Service Differentiation with Many Servers and Time-Varying Arrival Rates
Online Supplement to Delay-Based Service Differentiation with Many Servers and Time-Varying Arrival Rates Xu Sun and Ward Whitt Department of Industrial Engineering and Operations Research, Columbia University
More informationStabilizing Customer Abandonment in Many-Server Queues with Time-Varying Arrivals
Submitted to Operations Research manuscript OPRE-29-6-259; OPRE-211-12-597.R1 Stabilizing Customer Abandonment in Many-Server Queues with -Varying Arrivals Yunan Liu Department of Industrial Engineering,
More informationThe Performance Impact of Delay Announcements
The Performance Impact of Delay Announcements Taking Account of Customer Response IEOR 4615, Service Engineering, Professor Whitt Supplement to Lecture 21, April 21, 2015 Review: The Purpose of Delay Announcements
More informationStabilizing Performance in a Service System with Time-Varying Arrivals and Customer Feedback
Stabilizing Performance in a Service System with Time-Varying Arrivals and Customer Feedback Yunan Liu a, Ward Whitt b, a Department of Industrial Engineering, North Carolina State University, Raleigh,
More informationDELAY PREDICTORS FOR CUSTOMER SERVICE SYSTEMS WITH TIME-VARYING PARAMETERS. Rouba Ibrahim Ward Whitt
Proceedings of the 21 Winter Simulation Conference B. Johansson, S. Jain, J. Montoya-Torres, J. Hugan, and E. Yücesan, eds. DELAY PREDICTORS FOR CUSTOMER SERVICE SYSTEMS WITH TIME-VARYING PARAMETERS Rouba
More informationec1 e-companion to Liu and Whitt: Stabilizing Performance
ec1 This page is intentionally blank. Proper e-companion title page, with INFORMS branding and exact metadata of the main paper, will be produced by the INFORMS office when the issue is being assembled.
More informationLecture 9: Deterministic Fluid Models and Many-Server Heavy-Traffic Limits. IEOR 4615: Service Engineering Professor Whitt February 19, 2015
Lecture 9: Deterministic Fluid Models and Many-Server Heavy-Traffic Limits IEOR 4615: Service Engineering Professor Whitt February 19, 2015 Outline Deterministic Fluid Models Directly From Data: Cumulative
More informationElectronic Companion Fluid Models for Overloaded Multi-Class Many-Server Queueing Systems with FCFS Routing
Submitted to Management Science manuscript MS-251-27 Electronic Companion Fluid Models for Overloaded Multi-Class Many-Server Queueing Systems with FCFS Routing Rishi Talreja, Ward Whitt Department of
More informationAPPENDIX to Stabilizing Performance in Many-Server Queues with Time-Varying Arrivals and Customer Feedback
APPENDIX to Stabilizing Performance in Many-Server Queues with -Varying Arrivals and Customer Feedback Yunan Liu and Ward Whitt Department of Industrial Engineering North Carolina State University Raleigh,
More informationDynamic Control of Parallel-Server Systems
Dynamic Control of Parallel-Server Systems Jim Dai Georgia Institute of Technology Tolga Tezcan University of Illinois at Urbana-Champaign May 13, 2009 Jim Dai (Georgia Tech) Many-Server Asymptotic Optimality
More informationThe G t /GI/s t +GI many-server fluid queue. Yunan Liu & Ward Whitt. Queueing Systems Theory and Applications. ISSN Volume 71 Number 4
The G t /GI/s t +GI many-server fluid queue Yunan Liu & Ward Whitt Queueing Systems Theory and Applications ISSN 257-13 Volume 71 Number 4 Queueing Syst (212) 71:45-444 DOI 1.17/s11134-12-9291-1 23 Your
More informationSTABILIZING PERFORMANCE IN NETWORKS OF QUEUES WITH TIME-VARYING ARRIVAL RATES
Probability in the Engineering and Informational Sciences, 28, 24, 49 449. doi:.7/s2699648484 STABILIZING PERFORMANCE IN NETWORKS OF QUEUES WITH TIME-VARYING ARRIVAL RATES YUNAN LIU AND WARD WHITT Department
More informationarxiv: v1 [math.pr] 16 Jan 2014
The Annals of Applied Probability 214, Vol. 24, No. 1, 378 421 DOI: 1.1214/13-AAP927 c Institute of Mathematical Statistics, 214 arxiv:141.3933v1 [math.pr] 16 Jan 214 MANY-SERVER HEAVY-TRAFFIC LIMIT FOR
More informationMany-Server Loss Models with Non-Poisson Time-Varying Arrivals
Many-Server Loss Models with Non-Poisson Time-Varying Arrivals Ward Whitt, Jingtong Zhao Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027 Received
More informationENGINEERING SOLUTION OF A BASIC CALL-CENTER MODEL
ENGINEERING SOLUTION OF A BASIC CALL-CENTER MODEL by Ward Whitt Department of Industrial Engineering and Operations Research Columbia University, New York, NY 10027 Abstract An algorithm is developed to
More informationAuthor's personal copy
Queueing Syst (215) 81:341 378 DOI 1.17/s11134-15-9462-x Stabilizing performance in a single-server queue with time-varying arrival rate Ward Whitt 1 Received: 5 July 214 / Revised: 7 May 215 / Published
More information(b) What is the variance of the time until the second customer arrives, starting empty, assuming that we measure time in minutes?
IEOR 3106: Introduction to Operations Research: Stochastic Models Fall 2006, Professor Whitt SOLUTIONS to Final Exam Chapters 4-7 and 10 in Ross, Tuesday, December 19, 4:10pm-7:00pm Open Book: but only
More informationDelay Announcements. Predicting Queueing Delays for Delay Announcements. IEOR 4615, Service Engineering, Professor Whitt. Lecture 21, April 21, 2015
Delay Announcements Predicting Queueing Delays for Delay Announcements IEOR 4615, Service Engineering, Professor Whitt Lecture 21, April 21, 2015 OUTLINE Delay Announcements: Why, What and When? [Slide
More informationREAL-TIME DELAY ESTIMATION BASED ON DELAY HISTORY IN MANY-SERVER SERVICE SYSTEMS WITH TIME-VARYING ARRIVALS
REAL-TIME DELAY ESTIMATION BASED ON DELAY HISTORY IN MANY-SERVER SERVICE SYSTEMS WITH TIME-VARYING ARRIVALS by Rouba Ibrahim and Ward Whitt IEOR Department Columbia University 500 West 120th Street 313
More informationWhat You Should Know About Queueing Models To Set Staffing Requirements in Service Systems
What You Should Know About Queueing Models To Set Staffing Requirements in Service Systems by Ward Whitt Department of Industrial Engineering and Operations Research Columbia University 304 S. W. Mudd
More informationMODELING WEBCHAT SERVICE CENTER WITH MANY LPS SERVERS
MODELING WEBCHAT SERVICE CENTER WITH MANY LPS SERVERS Jiheng Zhang Oct 26, 211 Model and Motivation Server Pool with multiple LPS servers LPS Server K Arrival Buffer. Model and Motivation Server Pool with
More informationA Fluid Approximation for Service Systems Responding to Unexpected Overloads
OPERATIONS RESEARCH Vol. 59, No. 5, September October 2011, pp. 1159 1170 issn 0030-364X eissn 1526-5463 11 5905 1159 http://dx.doi.org/10.1287/opre.1110.0985 2011 INFORMS A Fluid Approximation for Service
More informationDesigning a Telephone Call Center with Impatient Customers
Designing a Telephone Call Center with Impatient Customers with Ofer Garnett Marty Reiman Sergey Zeltyn Appendix: Strong Approximations of M/M/ + Source: ErlangA QEDandSTROG FIAL.tex M/M/ + M System Poisson
More informationREAL-TIME DELAY ESTIMATION BASED ON DELAY HISTORY IN MANY-SERVER SERVICE SYSTEMS WITH TIME-VARYING ARRIVALS
REAL-TIME DELAY ESTIMATION BASED ON DELAY HISTORY IN MANY-SERVER SERVICE SYSTEMS WITH TIME-VARYING ARRIVALS Abstract Motivated by interest in making delay announcements in service systems, we study real-time
More informationControl of Fork-Join Networks in Heavy-Traffic
in Heavy-Traffic Asaf Zviran Based on MSc work under the guidance of Rami Atar (Technion) and Avishai Mandelbaum (Technion) Industrial Engineering and Management Technion June 2010 Introduction Network
More informationEconomy of Scale in Multiserver Service Systems: A Retrospective. Ward Whitt. IEOR Department. Columbia University
Economy of Scale in Multiserver Service Systems: A Retrospective Ward Whitt IEOR Department Columbia University Ancient Relics A. K. Erlang (1924) On the rational determination of the number of circuits.
More informationClass 11 Non-Parametric Models of a Service System; GI/GI/1, GI/GI/n: Exact & Approximate Analysis.
Service Engineering Class 11 Non-Parametric Models of a Service System; GI/GI/1, GI/GI/n: Exact & Approximate Analysis. G/G/1 Queue: Virtual Waiting Time (Unfinished Work). GI/GI/1: Lindley s Equations
More informationWait-Time Predictors for Customer Service Systems With Time-Varying Demand and Capacity
Submitted to Operations Research manuscript (Please, provide the mansucript number!) Wait-Time Predictors for Customer Service Systems With Time-Varying Demand and Capacity Rouba Ibrahim, Ward Whitt Department
More informationOverflow Networks: Approximations and Implications to Call-Center Outsourcing
Overflow Networks: Approximations and Implications to Call-Center Outsourcing Itai Gurvich (Northwestern University) Joint work with Ohad Perry (CWI) Call Centers with Overflow λ 1 λ 2 Source of complexity:
More informationESTIMATING WAITING TIMES WITH THE TIME-VARYING LITTLE S LAW
ESTIMATING WAITING TIMES WITH THE TIME-VARYING LITTLE S LAW Song-Hee Kim and Ward Whitt Industrial Engineering and Operations Research Columbia University New York, NY, 10027 {sk3116, ww2040}@columbia.edu
More informationStochastic Networks and Parameter Uncertainty
Stochastic Networks and Parameter Uncertainty Assaf Zeevi Graduate School of Business Columbia University Stochastic Processing Networks Conference, August 2009 based on joint work with Mike Harrison Achal
More informationWait-Time Predictors for Customer Service Systems with Time-Varying Demand and Capacity
OPERATIONS RESEARCH Vol. 59, No. 5, September October 211, pp. 116 1118 issn 3-364X eissn 1526-5463 11 595 116 http://dx.doi.org/1.1287/opre.111.974 211 INFORMS Wait-Time Predictors for Customer Service
More informationQueueing Theory II. Summary. ! M/M/1 Output process. ! Networks of Queue! Method of Stages. ! General Distributions
Queueing Theory II Summary! M/M/1 Output process! Networks of Queue! Method of Stages " Erlang Distribution " Hyperexponential Distribution! General Distributions " Embedded Markov Chains M/M/1 Output
More informationHEAVY-TRAFFIC LIMITS FOR STATIONARY NETWORK FLOWS. By Ward Whitt, and Wei You November 28, 2018
Stochastic Systems HEAVY-TRAFFIC LIMITS FOR STATIONARY NETWORK FLOWS By Ward Whitt, and Wei You November 28, 2018 We establish heavy-traffic limits for the stationary flows in generalized Jackson networks,
More informationMarkovian N-Server Queues (Birth & Death Models)
Markovian -Server Queues (Birth & Death Moels) - Busy Perio Arrivals Poisson (λ) ; Services exp(µ) (E(S) = /µ) Servers statistically ientical, serving FCFS. Offere loa R = λ E(S) = λ/µ Erlangs Q(t) = number
More informationA STAFFING ALGORITHM FOR CALL CENTERS WITH SKILL-BASED ROUTING: SUPPLEMENTARY MATERIAL
A STAFFING ALGORITHM FOR CALL CENTERS WITH SKILL-BASED ROUTING: SUPPLEMENTARY MATERIAL by Rodney B. Wallace IBM and The George Washington University rodney.wallace@us.ibm.com Ward Whitt Columbia University
More informationAsymptotic Coupling of an SPDE, with Applications to Many-Server Queues
Asymptotic Coupling of an SPDE, with Applications to Many-Server Queues Mohammadreza Aghajani joint work with Kavita Ramanan Brown University March 2014 Mohammadreza Aghajanijoint work Asymptotic with
More informationA Diffusion Approximation for the G/GI/n/m Queue
OPERATIONS RESEARCH Vol. 52, No. 6, November December 2004, pp. 922 941 issn 0030-364X eissn 1526-5463 04 5206 0922 informs doi 10.1287/opre.1040.0136 2004 INFORMS A Diffusion Approximation for the G/GI/n/m
More informationSTAFFING A CALL CENTER WITH UNCERTAIN ARRIVAL RATE AND ABSENTEEISM
STAFFING A CALL CENTER WITH UNCERTAIN ARRIVAL RATE AND ABSENTEEISM by Ward Whitt Department of Industrial Engineering and Operations Research Columbia University, New York, NY 10027 6699 Abstract This
More informationA Diffusion Approximation for Stationary Distribution of Many-Server Queueing System In Halfin-Whitt Regime
A Diffusion Approximation for Stationary Distribution of Many-Server Queueing System In Halfin-Whitt Regime Mohammadreza Aghajani joint work with Kavita Ramanan Brown University APS Conference, Istanbul,
More informationSERVER STAFFING TO MEET TIME-VARYING DEMAND
SERVER STAFFING TO MEET TIME-VARYING DEMAND by Otis B. Jennings, 1 Avishai Mandelbaum, 2 William A. Massey 3 and Ward Whitt 4 AT&T Bell Laboratories September 12, 1994 Revision: July 11, 1995 1 School
More informationErlang-C = M/M/N. agents. queue ACD. arrivals. busy ACD. queue. abandonment BACK FRONT. lost calls. arrivals. lost calls
Erlang-C = M/M/N agents arrivals ACD queue Erlang-A lost calls FRONT BACK arrivals busy ACD queue abandonment lost calls Erlang-C = M/M/N agents arrivals ACD queue Rough Performance Analysis
More informationStochastic process. X, a series of random variables indexed by t
Stochastic process X, a series of random variables indexed by t X={X(t), t 0} is a continuous time stochastic process X={X(t), t=0,1, } is a discrete time stochastic process X(t) is the state at time t,
More informationQ = (c) Assuming that Ricoh has been working continuously for 7 days, what is the probability that it will remain working at least 8 more days?
IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2005, Professor Whitt, Second Midterm Exam Chapters 5-6 in Ross, Thursday, March 31, 11:00am-1:00pm Open Book: but only the Ross
More informationIEOR 8100: Topics in OR: Asymptotic Methods in Queueing Theory. Fall 2009, Professor Whitt. Class Lecture Notes: Wednesday, September 9.
IEOR 8100: Topics in OR: Asymptotic Methods in Queueing Theory Fall 2009, Professor Whitt Class Lecture Notes: Wednesday, September 9. Heavy-Traffic Limits for the GI/G/1 Queue 1. The GI/G/1 Queue We will
More informationTOWARDS BETTER MULTI-CLASS PARAMETRIC-DECOMPOSITION APPROXIMATIONS FOR OPEN QUEUEING NETWORKS
TOWARDS BETTER MULTI-CLASS PARAMETRIC-DECOMPOSITION APPROXIMATIONS FOR OPEN QUEUEING NETWORKS by Ward Whitt AT&T Bell Laboratories Murray Hill, NJ 07974-0636 March 31, 199 Revision: November 9, 199 ABSTRACT
More informationIEOR 6711: Stochastic Models I, Fall 2003, Professor Whitt. Solutions to Final Exam: Thursday, December 18.
IEOR 6711: Stochastic Models I, Fall 23, Professor Whitt Solutions to Final Exam: Thursday, December 18. Below are six questions with several parts. Do as much as you can. Show your work. 1. Two-Pump Gas
More informationSENSITIVITY TO THE SERVICE-TIME DISTRIBUTION IN THE NONSTATIONARY ERLANG LOSS MODEL
SENSITIVITY TO THE SERVICE-TIME DISTRIBUTION IN THE NONSTATIONARY ERLANG LOSS MODEL by Jimmie L. Davis, 1 William A. Massey 2 and Ward Whitt 3 AT&T Bell Laboratories October 20, 1992 Revision: May 17,
More informationThis paper investigates the impact of dependence among successive service times on the transient and
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol. 14, No. 2, Spring 212, pp. 262 278 ISSN 1523-4614 (print) ISSN 1526-5498 (online) http://dx.doi.org/1.1287/msom.111.363 212 INFORMS The Impact of Dependent
More informationA Fluid Approximation for Service Systems Responding to Unexpected Overloads
Submitted to Operations Research manuscript 28-9-55.R2, Longer Online Version 6/28/1 A Fluid Approximation for Service Systems Responding to Unexpected Overloads Ohad Perry Centrum Wiskunde & Informatica
More informationMotivated by models of tenant assignment in public housing, we study approximating deterministic fluid
MANAGEMENT SCIENCE Vol. 54, No. 8, August 2008, pp. 1513 1527 issn 0025-1909 eissn 1526-5501 08 5408 1513 informs doi 10.1287/mnsc.1080.0868 2008 INFORMS Fluid Models for Overloaded Multiclass Many-Server
More informationThe Offered-Load Process: Modeling, Inference and Applications. Research Thesis
The Offered-Load Process: Modeling, Inference and Applications Research Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Statistics Michael Reich Submitted
More informationSENSITIVITY OF PERFORMANCE IN THE ERLANG-A QUEUEING MODEL TO CHANGES IN THE MODEL PARAMETERS
SENSITIVITY OF PERFORMANCE IN THE ERLANG-A QUEUEING MODEL TO CHANGES IN THE MODEL PARAMETERS by Ward Whitt Department of Industrial Engineering and Operations Research Columbia University, New York, NY
More informationA DIFFUSION APPROXIMATION FOR THE G/GI/n/m QUEUE
A DIFFUSION APPROXIMATION FOR THE G/GI/n/m QUEUE by Ward Whitt Department of Industrial Engineering and Operations Research Columbia University, New York, NY 10027 July 5, 2002 Revision: June 27, 2003
More informationFigure 10.1: Recording when the event E occurs
10 Poisson Processes Let T R be an interval. A family of random variables {X(t) ; t T} is called a continuous time stochastic process. We often consider T = [0, 1] and T = [0, ). As X(t) is a random variable
More informationStochastic grey-box modeling of queueing systems: fitting birth-and-death processes to data
Queueing Syst (2015) 79:391 426 DOI 10.1007/s11134-014-9429-3 Stochastic grey-box modeling of queueing systems: fitting birth-and-death processes to data James Dong Ward Whitt Received: 8 January 2014
More informationQUEUING SYSTEM. Yetunde Folajimi, PhD
QUEUING SYSTEM Yetunde Folajimi, PhD Part 2 Queuing Models Queueing models are constructed so that queue lengths and waiting times can be predicted They help us to understand and quantify the effect of
More informationRouting and Staffing in Large-Scale Service Systems: The Case of Homogeneous Impatient Customers and Heterogeneous Servers 1
Routing and Staffing in Large-Scale Service Systems: The Case of Homogeneous Impatient Customers and Heterogeneous Servers 1 Mor Armony 2 Avishai Mandelbaum 3 June 25, 2008 Abstract Motivated by call centers,
More informationUSING DIFFERENT RESPONSE-TIME REQUIREMENTS TO SMOOTH TIME-VARYING DEMAND FOR SERVICE
USING DIFFERENT RESPONSE-TIME REQUIREMENTS TO SMOOTH TIME-VARYING DEMAND FOR SERVICE by Ward Whitt 1 AT&T Labs September 9, 1997 Revision: November 20, 1998 Operations Research Letters 24 (1999) 1 10 1
More informationQueues with Many Servers and Impatient Customers
MATHEMATICS OF OPERATIOS RESEARCH Vol. 37, o. 1, February 212, pp. 41 65 ISS 364-765X (print) ISS 1526-5471 (online) http://dx.doi.org/1.1287/moor.111.53 212 IFORMS Queues with Many Servers and Impatient
More informationDesign and evaluation of overloaded service systems with skill based routing, under FCFS policies
Design and evaluation of overloaded service systems with skill based routing, under FCFS policies Ivo Adan Marko Boon Gideon Weiss April 2, 2013 Abstract We study an overloaded service system with servers
More informationIntroduction to Markov Chains, Queuing Theory, and Network Performance
Introduction to Markov Chains, Queuing Theory, and Network Performance Marceau Coupechoux Telecom ParisTech, departement Informatique et Réseaux marceau.coupechoux@telecom-paristech.fr IT.2403 Modélisation
More informationNon Markovian Queues (contd.)
MODULE 7: RENEWAL PROCESSES 29 Lecture 5 Non Markovian Queues (contd) For the case where the service time is constant, V ar(b) = 0, then the P-K formula for M/D/ queue reduces to L s = ρ + ρ 2 2( ρ) where
More informationNew Perspectives on the Erlang-A Queue
New Perspectives on the Erlang-A Queue arxiv:1712.8445v1 [math.pr] 22 Dec 217 Andrew Daw School of Operations Research and Information Engineering Cornell University 257 Rhodes Hall, Ithaca, NY 14853 amd399@cornell.edu
More informationQueues and Queueing Networks
Queues and Queueing Networks Sanjay K. Bose Dept. of EEE, IITG Copyright 2015, Sanjay K. Bose 1 Introduction to Queueing Models and Queueing Analysis Copyright 2015, Sanjay K. Bose 2 Model of a Queue Arrivals
More informationA Robust Queueing Network Analyzer Based on Indices of Dispersion
A Robust Queueing Network Analyzer Based on Indices of Dispersion Wei You (joint work with Ward Whitt) Columbia University INFORMS 2018, Phoenix November 6, 2018 1/20 Motivation Many complex service systems
More informationProbability and Statistics Concepts
University of Central Florida Computer Science Division COT 5611 - Operating Systems. Spring 014 - dcm Probability and Statistics Concepts Random Variable: a rule that assigns a numerical value to each
More informationLarge-time asymptotics for the G t /M t /s t + GI t many-server fluid queue with abandonment
Queueing Syst (211) 67: 145 182 DOI 1.17/s11134-1-928-8 Large-time asymptotics for the G t /M t /s t + GI t many-server fluid queue with abandonment Yunan Liu Ward Whitt Received: 12 March 21 / Revised:
More informationISE/OR 762 Stochastic Simulation Techniques
ISE/OR 762 Stochastic Simulation Techniques Topic 0: Introduction to Discrete Event Simulation Yunan Liu Department of Industrial and Systems Engineering NC State University January 9, 2018 Yunan Liu (NC
More informationDynamic Call Center Routing Policies Using Call Waiting and Agent Idle Times Online Supplement
Submitted to imanufacturing & Service Operations Management manuscript MSOM-11-370.R3 Dynamic Call Center Routing Policies Using Call Waiting and Agent Idle Times Online Supplement (Authors names blinded
More informationState Space Collapse in Many-Server Diffusion Limits of Parallel Server Systems. J. G. Dai. Tolga Tezcan
State Space Collapse in Many-Server Diffusion imits of Parallel Server Systems J. G. Dai H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia
More informationStochastic Models in Computer Science A Tutorial
Stochastic Models in Computer Science A Tutorial Dr. Snehanshu Saha Department of Computer Science PESIT BSC, Bengaluru WCI 2015 - August 10 to August 13 1 Introduction 2 Random Variable 3 Introduction
More informationPerformance Evaluation of Queuing Systems
Performance Evaluation of Queuing Systems Introduction to Queuing Systems System Performance Measures & Little s Law Equilibrium Solution of Birth-Death Processes Analysis of Single-Station Queuing Systems
More informationPart I Stochastic variables and Markov chains
Part I Stochastic variables and Markov chains Random variables describe the behaviour of a phenomenon independent of any specific sample space Distribution function (cdf, cumulative distribution function)
More informationHITTING TIME IN AN ERLANG LOSS SYSTEM
Probability in the Engineering and Informational Sciences, 16, 2002, 167 184+ Printed in the U+S+A+ HITTING TIME IN AN ERLANG LOSS SYSTEM SHELDON M. ROSS Department of Industrial Engineering and Operations
More informationSlides 9: Queuing Models
Slides 9: Queuing Models Purpose Simulation is often used in the analysis of queuing models. A simple but typical queuing model is: Queuing models provide the analyst with a powerful tool for designing
More informationDynamic Control of a Tandem Queueing System with Abandonments
Dynamic Control of a Tandem Queueing System with Abandonments Gabriel Zayas-Cabán 1 Jungui Xie 2 Linda V. Green 3 Mark E. Lewis 1 1 Cornell University Ithaca, NY 2 University of Science and Technology
More informationIntro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks. Queuing Networks. Florence Perronnin
Queuing Networks Florence Perronnin Polytech Grenoble - UGA March 23, 27 F. Perronnin (UGA) Queuing Networks March 23, 27 / 46 Outline Introduction to Queuing Networks 2 Refresher: M/M/ queue 3 Reversibility
More informationThe M/M/n+G Queue: Summary of Performance Measures
The M/M/n+G Queue: Summary of Performance Measures Avishai Mandelbaum and Sergey Zeltyn Faculty of Industrial Engineering & Management Technion Haifa 32 ISRAEL emails: avim@tx.technion.ac.il zeltyn@ie.technion.ac.il
More informationPart II: continuous time Markov chain (CTMC)
Part II: continuous time Markov chain (CTMC) Continuous time discrete state Markov process Definition (Markovian property) X(t) is a CTMC, if for any n and any sequence t 1
More informationIntroduction to Queueing Theory
Introduction to Queueing Theory Raj Jain Washington University in Saint Louis Jain@eecs.berkeley.edu or Jain@wustl.edu A Mini-Course offered at UC Berkeley, Sept-Oct 2012 These slides and audio/video recordings
More informationQueueing Theory. VK Room: M Last updated: October 17, 2013.
Queueing Theory VK Room: M1.30 knightva@cf.ac.uk www.vincent-knight.com Last updated: October 17, 2013. 1 / 63 Overview Description of Queueing Processes The Single Server Markovian Queue Multi Server
More informationCONTINUITY OF A QUEUEING INTEGRAL REPRESENTATION IN THE M 1 TOPOLOGY. Department of Industrial Engineering and Operations Research Columbia University
Submitted to the Annals of Applied Probability CONTINUITY OF A QUEUEING INTEGRAL REPRESENTATION IN THE M 1 TOPOLOGY By Guodong Pang and Ward Whitt Department of Industrial Engineering and Operations Research
More informationFluid Limit of A Many-Server Queueing Network with Abandonment
Fluid Limit of A Many-Server Queueing Networ with Abandonment Weining Kang Guodong Pang October 22, 213 Abstract This paper studies a non-marovian many-server queueing networ with abandonment, where externally
More informationS n = x + X 1 + X X n.
0 Lecture 0 0. Gambler Ruin Problem Let X be a payoff if a coin toss game such that P(X = ) = P(X = ) = /2. Suppose you start with x dollars and play the game n times. Let X,X 2,...,X n be payoffs in each
More informationPBW 654 Applied Statistics - I Urban Operations Research
PBW 654 Applied Statistics - I Urban Operations Research Lecture 2.I Queuing Systems An Introduction Operations Research Models Deterministic Models Linear Programming Integer Programming Network Optimization
More informationTHE HEAVY-TRAFFIC BOTTLENECK PHENOMENON IN OPEN QUEUEING NETWORKS. S. Suresh and W. Whitt AT&T Bell Laboratories Murray Hill, New Jersey 07974
THE HEAVY-TRAFFIC BOTTLENECK PHENOMENON IN OPEN QUEUEING NETWORKS by S. Suresh and W. Whitt AT&T Bell Laboratories Murray Hill, New Jersey 07974 ABSTRACT This note describes a simulation experiment involving
More informationQueueing Theory and Simulation. Introduction
Queueing Theory and Simulation Based on the slides of Dr. Dharma P. Agrawal, University of Cincinnati and Dr. Hiroyuki Ohsaki Graduate School of Information Science & Technology, Osaka University, Japan
More informationProactive Care with Degrading Class Types
Proactive Care with Degrading Class Types Yue Hu (DRO, Columbia Business School) Joint work with Prof. Carri Chan (DRO, Columbia Business School) and Prof. Jing Dong (DRO, Columbia Business School) Motivation
More informationSince D has an exponential distribution, E[D] = 0.09 years. Since {A(t) : t 0} is a Poisson process with rate λ = 10, 000, A(0.
IEOR 46: Introduction to Operations Research: Stochastic Models Chapters 5-6 in Ross, Thursday, April, 4:5-5:35pm SOLUTIONS to Second Midterm Exam, Spring 9, Open Book: but only the Ross textbook, the
More informationM/G/1 and M/G/1/K systems
M/G/1 and M/G/1/K systems Dmitri A. Moltchanov dmitri.moltchanov@tut.fi http://www.cs.tut.fi/kurssit/elt-53606/ OUTLINE: Description of M/G/1 system; Methods of analysis; Residual life approach; Imbedded
More informationNEW FRONTIERS IN APPLIED PROBABILITY
J. Appl. Prob. Spec. Vol. 48A, 209 213 (2011) Applied Probability Trust 2011 NEW FRONTIERS IN APPLIED PROBABILITY A Festschrift for SØREN ASMUSSEN Edited by P. GLYNN, T. MIKOSCH and T. ROLSKI Part 4. Simulation
More informationOther properties of M M 1
Other properties of M M 1 Přemysl Bejda premyslbejda@gmail.com 2012 Contents 1 Reflected Lévy Process 2 Time dependent properties of M M 1 3 Waiting times and queue disciplines in M M 1 Contents 1 Reflected
More informationFork-Join Networks in Heavy Traffic: Diffusion Approximations and Control. M.Sc. Research Proposal
Fork-Join Networks in Heavy Traffic: Diffusion Approximations and Control M.Sc. Research Proposal Asaf Zviran Advisors: Prof. Rami Atar Prof. Avishai Mandelbaum Faculty of Industrial Engineering and Management
More informationStochastic-Process Limits
Ward Whitt Stochastic-Process Limits An Introduction to Stochastic-Process Limits and Their Application to Queues With 68 Illustrations Springer Contents Preface vii 1 Experiencing Statistical Regularity
More informationEstimation of arrival and service rates for M/M/c queue system
Estimation of arrival and service rates for M/M/c queue system Katarína Starinská starinskak@gmail.com Charles University Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics
More information