Transcription
S TAFFING AND C ONTROL OF I NSTANTM ESSAGING BASED C USTOMER S ERVICEC ENTERSJiheng Zhang(Joint work with Jun Luo)
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsModel and MotivationServer Pool with multiple LPS serversLPS Server KArrivalBuffer.Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsModel and MotivationServer Pool with multiple LPS serversLPS Server KArrivalBuffer.Customer contact centers via instant messaging.Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleModelingServers work at different speed depending how many customerin service.210service rate34Service Rate / 0.5 hour12345number of customers in serviceInstant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleModelingServers work at different speed depending how many customerin service.201service rate34Service Rate / 0.5 hour12345number of customers in serviceClassify the pool of N homogeneous servers into “levels”.Level k: all servers serving k customers.Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleModelingThe classical -model:12µ1µ2.K12µKµ1µ2Instant Messaging based Services Centers / J. Zhang.KµK
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleModelingThe classical -model:.K1 1 2 2Kµ1 µ1 µ2 µ2µK µK11 2.K2 .µ1 µ1µ2 µ2Instant Messaging based Services Centers / J. ZhangKµK µK
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleLiterature ReviewMany-server QueuesPuhalskii 2007, Mandelbaum, Massey & Reiman 1998, . . .Perry & Whitt 2010 – now. . .Averaging PrincipleCoffman, Puhalskii & Reiman 1995, 1998Kurtz 1992, Hunt & Kurtz 1994, . . .LPS QueuesZhang, Dai and Zwart 2010, 2011Zhang & Zwart 2008, Gupta & Zhang 2011, . . .Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsNotationsServer pool Z(t) (Z0 (t), Z1 (t), . . . , ZK (t)) NK 1Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsNotationsServer pool Z(t) (Z0 (t), Z1 (t), . . . , ZK (t)) NK 1Queue Q(t) NInstant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsNotationsServer pool Z(t) (Z0 (t), Z1 (t), . . . , ZK (t)) NK 1Queue Q(t) NKXZk (t) N,and Q(t)(N ZK (t)) 0.k 0Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsNotationsServer pool Z(t) (Z0 (t), Z1 (t), . . . , ZK (t)) NK 1Queue Q(t) NKXZk (t) N,and Q(t)(N ZK (t)) 0.k 0Service Z t Dk (t) Sk γkZk (s)ds0Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsNotationsServer pool Z(t) (Z0 (t), Z1 (t), . . . , ZK (t)) NK 1Queue Q(t) NKXZk (t) N,and Q(t)(N ZK (t)) 0.k 0Service Z t Dk (t) Sk γkZk (s)ds0Key assumption: exponential service timeInstant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsNotationsServer pool Z(t) (Z0 (t), Z1 (t), . . . , ZK (t)) NK 1Queue Q(t) NKXZk (t) N,and Q(t)(N ZK (t)) 0.k 0Service Z t Dk (t) Sk γkZk (s)ds0Key assumption: exponential service timeThe index (routing)i (t) min{0 k K : Zk (t) 0}.Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleSystem Dynamics – some simulationsnOne Sample Path of Stochastic Process Z (t)200ZHeadcount0Z1501Z2Z1003Z4500051015TimeOne Sample Path of Index Processin(t)*Index32100510Timeλn 400, N n 200, K 6 and γ (1, 1.6, 1.8, 2.2, 2.3, 2.4).Instant Messaging based Services Centers / J. Zhang15
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsSystem DynamicsWhen will zk (s) jump by 1Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsSystem DynamicsWhen will zk (s) jump by 1i (s ) k 1, and an arrival happens at sInstant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsSystem DynamicsWhen will zk (s) jump by 1i (s ) k 1, and an arrival happens at sa service completion from group k 1 at sInstant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsSystem DynamicsWhen will zk (s) jump by 1i (s ) k 1, and an arrival happens at sa service completion from group k 1 at sWhen will zk (s) jump by -1Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsSystem DynamicsWhen will zk (s) jump by 1i (s ) k 1, and an arrival happens at sa service completion from group k 1 at sWhen will zk (s) jump by -1i (s ) k, and an arrival happens at sInstant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsSystem DynamicsWhen will zk (s) jump by 1i (s ) k 1, and an arrival happens at sa service completion from group k 1 at sWhen will zk (s) jump by -1i (s ) k, and an arrival happens at sa service completion from group k at sInstant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleSystem DynamicsWhen will zk (s) jump by 1i (s ) k 1, and an arrival happens at sa service completion from group k 1 at sWhen will zk (s) jump by -1i (s ) k, and an arrival happens at sa service completion from group k at sDynamic Equation for Zk , 0 k KZ tZ tZk (t) Zk (0) 1{i (s ) k 1} dΛ(s) 1{Q(s ) 0} dDk 1 (s)00Z t 1{i (s ) k} dΛ(s) Dk (t)0Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsSystem DynamicsDynamic Equation for Z0ZZ0 (t) Z0 (0) t1{i (s ) 0} dΛ(s) D1 (t)0Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleSystem DynamicsDynamic Equation for Z0ZZ0 (t) Z0 (0) t1{i (s ) 0} dΛ(s) D1 (t)0Dynamic Equation for ZKZ tZ tZK (t) ZK (0) 1{i (s ) K 1} dΛ(s) 1{Q(s ) 0} dDK (s)00Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleSystem DynamicsDynamic Equation for Z0ZZ0 (t) Z0 (0) t1{i (s ) 0} dΛ(s) D1 (t)0Dynamic Equation for ZKZ tZ tZK (t) ZK (0) 1{i (s ) K 1} dΛ(s) 1{Q(s ) 0} dDK (s)00Dynamic Equation for QZ tZ tQ(t) Q(0) 1{i (s ) K} dΛ(s) 1{Q(s) 0} dDK (s)00Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleConstant Arrival v.s. Time-Varying ArrivalArrival process, Λ(t), is assumed to be non-homogeneousPoisson process with rate λ(t).481216202404812162094.5arrival rateArrival Rate/ 0.5 Hour on Wednesday0 35.44174.5arrival rate02404812162024Arrival Rate/ 0.5 Hour on ThursdayArrival Rate/ 0.5 Hour on FridayArrival Rate/ 0.5 Hour on Weekdays81216Time of the Day2024arrival rate0481216200 58.91arrival rate40 58.31660157.1Time of the Day155.5Time of the Day132Time of the Day0arrival rateArrival Rate/ 0.5 Hour on Tuesday0 65.44186.50 69.94arrival rateArrival Rate/ 0.5 Hour on Monday24Time of the DayInstant Messaging based Services Centers / J. Zhang0481216Time of the Day2024
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsA Heavy Traffic RegimeLarge number of servers to accomodate large demand.Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsA Heavy Traffic RegimeLarge number of servers to accomodate large demand.Consider a sequence of system indexed by n1 nN N,nand1 nλ (t) λ(t),nInstant Messaging based Services Centers / J. Zhangas n .Average Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleA Heavy Traffic RegimeLarge number of servers to accomodate large demand.Consider a sequence of system indexed by n1 n1 nN N, andλ (t) λ(t), as n .nnBut each server’s service rate γ (γ1 , . . . , γK ) is fixed.Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleA Heavy Traffic RegimeLarge number of servers to accomodate large demand.Consider a sequence of system indexed by n1 n1 nN N, andλ (t) λ(t), as n .nnBut each server’s service rate γ (γ1 , . . . , γK ) is fixed.Fluid Scaling:Z̄ n (t) Z n (t),nQ̄n (t) Qn (t).nInstant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsOptimalityUtility functionC̄Tn (N̄ n , K)1 cN̄ ETn ZT h Z̄ (s), Q̄ (s) ds .n0Instant Messaging based Services Centers / J. ZhangnAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsOptimalityUtility functionC̄Tn (N̄ n , K)1 cN̄ ETEg. linear cost h(z, q) n ZT h Z̄ (s), Q̄ (s) ds .n0X 1q kzk .λkInstant Messaging based Services Centers / J. ZhangnAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsOptimalityUtility functionC̄Tn (N̄ n , K)1 cN̄ ETEg. linear cost h(z, q) n ZT h Z̄ (s), Q̄ (s) ds .nn0X 1q kzk .λkStaffing {N̄ n } and control K is asymptotically optimal iflim sup C̄Tn (N̄ n , K ) lim inf C̄Tn (N̄ n , K).n n Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsOptimalityUtility functionC̄Tn (N̄ n , K)1 cN̄ ETEg. linear cost h(z, q) n ZT h Z̄ (s), Q̄ (s) ds .nn0X 1q kzk .λkStaffing {N̄ n } and control K is asymptotically optimal iflim sup C̄Tn (N̄ n , K ) lim inf C̄Tn (N̄ n , K).n n lim sup lim sup C̄Tn (N̄ n , K ) lim sup lim inf C̄Tn (N̄ n , K).T n T n Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsOptimal Staffing and Control - Finite HorizonP ROPOSITIONIn the heavy traffic regimeC̄Tn (N̄ n , K) CT (N, K),as n .Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleOptimal Staffing and Control - Finite Horizon(a) Arrival Rate Function(b) Cost Function3.238K 2K 3K 4363.1343Total CostArrival 1Number of Servers200.9TimeXγ (2, 3, 2.7, 3.2), c 19, h(z, q) 1 (kzk q)kInstant Messaging based Services Centers / J. Zhang1.05
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleOptimal Staffing and Control - Infinite HorizonP ROPOSITIONAssume λ(t) λ and γk is increasing and N λ/ sup γk . Fork K0any K K0lim lim C̄Tn (N̄ n , K) C(N),T n as n .Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleOptimal Staffing and Control - Infinite Horizon(a) Linear Holding Cost(b) Quadratic Holding Cost3.212Fluidn 50n 100n 2003.131110Cost2.9CostFluidn 50n 100n nt Messaging based Services Centers / J. Zhang0.60.7N0.80.91
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleOptimal Staffing and Control - Infinite Horizon(a) Linear Holding Cost(b) Quadratic Holding Cost3.212Fluidn 50n 100n 2003.131110Cost2.9CostFluidn 50n 100n 2002.82.7982.672.50.40.50.60.7N0.80.9610.40.5γ (1, 1.6, 1.9, 2.3, 2.6, 2.8)Xλ 1, c 2, and h(z, q) 1 (kzk q).kInstant Messaging based Services Centers / J. Zhang0.60.7N0.80.91
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleOptimal Staffing and Control - Infinite Horizon(a) Linear Holding Cost(b) Quadratic Holding Cost3.212Fluidn 50n 100n 2003.131110Cost2.9CostFluidn 50n 100n 2002.82.7982.672.50.40.50.60.7N0.80.9610.40.5γ (1, 1.8, 2.1, 2.2, 2.5, 2.7) Xλ 1, c 10, and h(z, q) (kzk q)2 .kInstant Messaging based Services Centers / J. Zhang0.60.7N0.80.91
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsFluid ModelLet (z, q) be the fluid counterpart of (Z̄ n , Q̄n ).Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleFluid ModelLet (z, q) be the fluid counterpart of (Z̄ n , Q̄n ).S {(z, q) [0, N]K 1 R :KXzk N and q(N zK ) 0}k 0Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleFluid ModelLet (z, q) be the fluid counterpart of (Z̄ n , Q̄n ).S {(z, q) [0, N]K 1 R :KXzk N and q(N zK ) 0}k 0ZHow do we deal witht1{i (s ) k} dΛ̄n (s)?0Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleFluid ModelLet (z, q) be the fluid counterpart of (Z̄ n , Q̄n ).S {(z, q) [0, N]K 1 R :KXzk N and q(N zK ) 0}k 0ZHow do we deal witht1{i (s ) k} dΛ̄n (s)?0Introduce the mapping f : RK 2 [0, 1]K 1 , γk 1 zk 1 1, if k I(z) 1, λ γzk k fk (z, λ) 1 , if k I(z), λ 0,otherwise,Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleFluid ModelLet (z, q) be the fluid counterpart of (Z̄ n , Q̄n ).S {(z, q) [0, N]K 1 R :KXzk N and q(N zK ) 0}k 0ZHow do we deal witht1{i (s ) k} dΛ̄n (s)?0Introduce the mapping f : RK 2 [0, 1]K 1 , γk 1 zk 1 1, if k I(z) 1, λ γzk k fk (z, λ) 1 , if k I(z), λ 0,otherwise,where I(z) min{0 k K : zk 0}.Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleFluid ModelDivide the space S into two regionsS {(z0 , . . . , zK , q) S : q 0} ,S0 {(z0 , . . . , zK , q) S : q 0} .The fluid model can be defined by ODE’s in the two regions.Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleFluid ModelDivide the space S into two regionsS {(z0 , . . . , zK , q) S : q 0} ,S0 {(z0 , . . . , zK , q) S : q 0} .The fluid model can be defined by ODE’s in the two regions.on S z0k (t) 0,0 k K,0q (t) λ(t) γK N.Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsFluid Modelon S0z00 (t) f0 (z(t), λ(t))λ(t) γ1 z1 (t),z0k (t) fk 1 (z(t), λ(t))λ(t) γk 1 zk 1 (t) fk (z(t), λ(t))λ(t) γk zk (t),z0K (t)00 k K, fK 1 (z(t), λ(t))λ(t) γK zK (t),q (t) fK (z(t), λ(t))λ(t).Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsFluid Modelon S0z00 (t) f0 (z(t), λ(t))λ(t) γ1 z1 (t),z0k (t) fk 1 (z(t), λ(t))λ(t) γk 1 zk 1 (t) fk (z(t), λ(t))λ(t) γk zk (t),z0K (t)00 k K, fK 1 (z(t), λ(t))λ(t) γK zK (t),q (t) fK (z(t), λ(t))λ(t).The ODEs can be written into a vector form,(z0 , q0 ) Ψ(t, z, q).Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleExistence and Uniqueness of Fluid Model SolutionT HEOREM (E XISTENCE AND U NIQUENESS )Assume that λ(t) is a piece-wise continuous function. Thereexists a unique solution to the fluid model, i.e., the ODEs, withinitial condition (z(0), q(0)) S.Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleExistence and Uniqueness of Fluid Model SolutionT HEOREM (E XISTENCE AND U NIQUENESS )Assume that λ(t) is a piece-wise continuous function. Thereexists a unique solution to the fluid model, i.e., the ODEs, withinitial condition (z(0), q(0)) S.For the solution (z, q), define the associated fluid cost as1CT (N, K) cN TZTh(z(s), q(s))ds.0Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsInvariant State of Fluid ModelWhen the arrival rate is constant, we can study the “steadstate” of the fluid model.Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsInvariant State of Fluid ModelWhen the arrival rate is constant, we can study the “steadstate” of the fluid model.Assumption: γk is increasing in k.Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsInvariant State of Fluid ModelWhen the arrival rate is constant, we can study the “steadstate” of the fluid model.Assumption: γk is increasing in k.Then there exists some k such that(λ γk 1 zk 1 γk zk ,N zk 1 zk .Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsInvariant State of Fluid ModelWhen the arrival rate is constant, we can study the “steadstate” of the fluid model.Assumption: γk is increasing in k.Then there exists some k such that(λ γk 1 zk 1 γk zk ,N zk 1 zk .This implies thatzk γk 1 N λ,γk 1 γk zk 1 λ γk N.γk 1 γk Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsInvariant State of Fluid ModelWhen the arrival rate is constant, we can study the “steadstate” of the fluid model.Assumption: γk is increasing in k.Then there exists some k such that(λ γk 1 zk 1 γk zk ,N zk 1 zk .This implies thatzk γk 1 N λ,γk 1 γk zk 1 λ γk N.γk 1 γk Define it to be the invariant state I.Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsInvariant State of Fluid ModelT HEOREM (C ONVERGENCE TO I NVARIANT S TATE )Given any initial value (z(0), q(0)) S the fluid modelconverges, fast, to the invariant state z̃(N).Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleInvariant State of Fluid ModelT HEOREM (C ONVERGENCE TO I NVARIANT S TATE )Given any initial value (z(0), q(0)) S the fluid modelconverges, fast, to the invariant state z̃(N).Based on this proposition, it is easy to see that the fluid cost lim CT (N, K) cN h(z̃(N), 0) C(N).T Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleFunctional Law of Large NumbersT HEOREM (FWLLN)If (Z̄ n (0), Q̄n (0)) (z0 , q0 ), then (Z̄ n , Q̄n ) converges weakly inthe heavy traffic regime to the fluid model solution (z, q) with(z(0), q(0)) (z0 , q0 ).Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleSimulated Stochastic Model and the Fluid ModelComparison with Fluid Approximation for Zn2(t)1n 50n 100n son with Fluid Approximation for Q (t)1n 50n 100n 200Fluid0.80.60.40.20024681012Timeγ (1, 1, 6, 1.8, 2.2), K 4 and λ(t) 2 1 sin(t).λn (t) nλ(t), n 50, 100, 200.Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleSimulated Stochastic Model and the Fluid ModelComparison with Fluid 304050Time60708090100γ (1, 1.6, 1.8, 2.2, 2.3, 2.4), K 6, n 200, Nn 200, λn 390.z̃ (0, 0, 0, 5/8, 3/8, 0, 0) and q̃ 0.Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleSimulation with general service time distributionsPerformanceLevel 0Level 1Level 2Level 3Level 4Level 5Level 6Sojourn TimeExponentialErlang-2LN(1, 4)0.0004 0.00080.0088 0.00321.7325 0.0201122.2821 0.371675.9753 0.38370.0010 0.000701.7287 0.00070.0003 0.00070.0084 0.00241.7174 0.0154122.2991 0.253275.9740 0.26490.0006 0.000301.7287 0.00040.0005 0.00100.0102 0.00591.7553 0.0310122.3772 0.448875.8561 0.46830.0007 0.000601.7283 0.0011Instant Messaging based Services Centers / J. 08–
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsThe Averaging PrincipleOn a small interval [t, t δ],Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsThe Averaging PrincipleOn a small interval [t, t δ],the number of arrivals routed to level kZ1 t δ1{in (s ) k} dΛ̄n (s)δ tInstant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsThe Averaging PrincipleOn a small interval [t, t δ],the number of arrivals routed to level kZ1 t δ1{in (s ) k} dΛ̄n (s)δ tthe amount of fluid routed to level kfk (z(t), λ(t))λ(t)Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsThe Averaging PrincipleOn a small interval [t, t δ],the number of arrivals routed to level kZ1 t δ1{in (s ) k} dΛ̄n (s)δ tthe amount of fluid routed to level kfk (z(t), λ(t))λ(t)The averaging principleZ1 t δlim lim1{in (s ) k} dΛ̄n (s) fk (z(t), λ(t))λ(t).δ 0 n δ tInstant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleThe Averaging PrincipleThe interplay here between the in (t) and Z̄ n (t):The process Z̄ n (t) evolves slowly and determines thetransition rates for in (t).The process in (t) evolves quickly and its “steady state”determines the evolution of Z̄ n (t).Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleThe Averaging PrincipleThe interplay here between the in (t) and Z̄ n (t):The process Z̄ n (t) evolves slowly and determines thetransition rates for in (t).The process in (t) evolves quickly and its “steady state”determines the evolution of Z̄ n (t).To see this intuitively,1δZtt δ1{in (s ) k} λds 1nδZ0nδ1{in (t s ) k} λds.nWhen n becomes large, what determines the above integral is·actually the “steady state” of the process in (t ).nInstant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleRandom Measure and Martingale RepresentationDefine the random measure ν n byZ tnν ([0, t] A) 1{Z n (s ) A} ds,0for any t 0 and subset A Z̄K .Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleRandom Measure and Martingale RepresentationDefine the random measure ν n byZ tnν ([0, t] A) 1{Z n (s ) A} ds,0for any t 0 and subset A Z̄K .Let Ak {z Z̄K : zk 0 and zj 0, j k}, then1{in (t ) k} 1{Z n (t ) Ak } .Instant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleRandom Measure and Martingale RepresentationDefine the random measure ν n byZ tnν ([0, t] A) 1{Z n (s ) A} ds,0for any t 0 and subset A Z̄K .Let Ak {z Z̄K : zk 0 and zj 0, j k}, then1{in (t ) k} 1{Z n (t ) Ak } .Martingale RepresentationZ tnnλ̄n (s)ds,M̄a (t) Λ̄ (t) 0 Z t Z t 1nnnnM̄k (t) Sk γkZk (s)ds γkZk (s)ds ,n00Instant Messaging based Services Centers / J. Zhangk 1, . . . , K.
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleRandom Measure and Martingale RepresentationZ̄kn (t) Z̄kn (0)Z 0t1{Z n (s ) Ak 1 } dM̄an (s)ZZ 0t1{Z n (s ) Ak } dM̄an (s)tn M̄kn (t) 1{Q̄n (s ) 0} dM̄k 1(s)0ZZnn λ̄ (s)ν (ds dy) [0,t] Ak 1Z tγkZ̄kn (s)ds0λ̄n (s)ν n (ds dy)[0,t] AkZ γk 1tn1{Q̄n (s ) 0} Z̄k 1(s)ds,0Instant Messaging based Services Centers / J. Zhang0 k K,
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsA Missing GapInterchange of Steady State and Heavy Traffic Limits(Z̄ n (t), Q̄n (t))t nn(Z̄ , Z̄ )Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsA Missing GapInterchange of Steady State and Heavy Traffic Limits(Z̄ n (t), Q̄n (t))t nn(Z̄ , Z̄ )n (z(t), q(t))Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsA Missing GapInterchange of Steady State and Heavy Traffic Limits(Z̄ n (t), Q̄n (t))t nn(Z̄ , Z̄ )n (z(t), q(t))t (z , q )Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsA Missing GapInterchange of Steady State and Heavy Traffic Limits(Z̄ n (t), Q̄n (t))t nn(Z̄ , Z̄ )n n (z(t), q(t))t (z , q )Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsAverage PrincipleFuture ResearchCustomer AbandonmentInefficient levelsInterchange of LimitsDynamic control on finite horizon with time-varying arrivalDiffusion approximationInstant Messaging based Services Centers / J. Zhang
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsQuestions?Instant Messaging based Services Centers / J. ZhangAverage Principle
BackgroundStochastic ModelStaffing & ControlFluid ModelApproximationsThank you!Instant Messaging based Services Centers / J. ZhangAverage Principle
Customer contact centers via instant messaging. Instant Messaging based Services Centers / J. Zhang Background Stochastic Model Staffing & Control Fluid Model Approximations Average Principle
