Model and Motivation Many-server queue buffer N Many-server - PowerPoint PPT Presentation

F LUID M ODELS OF M ANY - SERVER Q UEUES WITH A BANDONMENT Jiheng Zhang June 10, 2010

Background Stochastic Model Fluid Model Functional LLN Approximations Model and Motivation Many-server queue buffer N Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations Model and Motivation Many-server queue buffer N Motivation: Customer call centers and other services areas. Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations Model and Motivation Many-server queue with abandonment buffer N Motivation: Customer call centers and other services areas. Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations Many-server queue v.s. Single-server queue Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations Many-server queue v.s. Single-server queue Large scale: high demand , need for high capacity Single server queue: increase speed Many-server queue: increase number of servers Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations A Real World Challenge The service time is not exponentially distributed! Brown et. al. Statistical analysis of a telephone call center: a queueing-science perspective . JASA 2005 In this research Arrival process: general Service/patient time distribution: general Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations Literature Review Many-server Queues Halfin and Whitt 1981 ( M / M / N ) Puhalskii and Reiman 2000 ( G / Ph / N ) Jelenkovi´ c, Mandelbaum and Momˇ cilovi´ c 2004 ( G / D / N ) Whitt 2005 ( G / H ∗ 2 / n / m ) Garmarnik and Momˇ cilovi´ c 2007 ( G / La / N ) Reed 2007, Puhalskii and Reed 2008 ( G / G / N ) Mandelbaum and Momˇ cilovi´ c 2008 ( G / G / N ) Kaspi and Ramanan 2009, Kaspi 2009 ( G / G / N ) . . . . . . Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations Literature Review Many-server Queues with Abandonment Whitt 2004 ( M / M / N + M ) Zeltyn and Mandelbaum 2005 ( G / M / N + G ) Whitt 2006 ( G / G / N + G ) Puhalskii 2008 ( M t / M t / N t + M t ) Kang and Ramanan 2008 ( G / G / N + G ) Mandelbaum and Momˇ cilovi´ c 2009 ( G / G / N + G ) Dai, He and Tezcan 2009 ( G / Ph / N + G ) . . . . . . Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations System Dynamics – example with N = 2 3 t = 0 Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations System Dynamics – example with N = 2 3 t = 0 2 t = 1 arrival 5 Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations System Dynamics – example with N = 2 3 t = 0 2 t = 1 arrival 5 1 2 arrival t = 2 4 Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations System Dynamics – example with N = 2 3 t = 0 2 t = 1 arrival 5 1 2 arrival t = 2 4 2 departure t = 3 3 Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations System Dynamics – example with N = 2 3 t = 0 2 t = 1 arrival 5 1 2 arrival t = 2 4 2 departure t = 3 3 1 t = 4 2 Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations System Dynamics – example with N = 2 3 t = 0 2 t = 1 arrival 5 1 2 arrival t = 2 4 2 departure t = 3 3 1 t = 4 2 departure t = 5 1 Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations Measure-valued State Descriptor Server pool Z ( t )( C ) : # of customers in server with remaining service time in C ⊂ ( 0 , ∞ ) Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations Measure-valued State Descriptor Server pool Z ( t )( C ) : # of customers in server with remaining service time in C ⊂ ( 0 , ∞ ) Evolution Z ( t 0 + t )( C ) = Z ( t 0 )( C + t ) + . . . ( t 0 , t 0 + t ) . . . Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations Measure-valued State Descriptor Server pool Z ( t )( C ) : # of customers in server with remaining service time in C ⊂ ( 0 , ∞ ) Evolution Z ( t 0 + t )( C ) = Z ( t 0 )( C + t ) + . . . ( t 0 , t 0 + t ) . . . Richness Z ( t ) = Z ( t )(( 0 , ∞ )) Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations Measure-valued State Descriptor Server pool Z ( t )( C ) : # of customers in server with remaining service time in C ⊂ ( 0 , ∞ ) Evolution Z ( t 0 + t )( C ) = Z ( t 0 )( C + t ) + . . . ( t 0 , t 0 + t ) . . . Richness Z ( t ) = Z ( t )(( 0 , ∞ )) Literature Gromoll, Puha & Williams ’02, Puha &Williams ’02, Gromoll ’06 Gromoll & Kurk ’07, Gromoll, Robert & Zwart ’08, . . . Zhang, Dai & Zwart ’07, ’08, Zhang & Zwart ’08 Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations Measure-valued State Descriptor Virtual buffer R ( t )( C ) : # of customers in virtual buffer with remaining patient time in C ⊂ ( −∞ , ∞ ) Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations Measure-valued State Descriptor Virtual buffer R ( t )( C ) : # of customers in virtual buffer with remaining patient time in C ⊂ ( −∞ , ∞ ) Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations Measure-valued State Descriptor Virtual buffer R ( t )( C ) : # of customers in virtual buffer with remaining patient time in C ⊂ ( −∞ , ∞ ) Evolution 4 1 2 1 6 buffer Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations Measure-valued State Descriptor Virtual buffer R ( t )( C ) : # of customers in virtual buffer with remaining patient time in C ⊂ ( −∞ , ∞ ) Evolution 5 3 0 1 0 buffer Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations Measure-valued State Descriptor Virtual buffer R ( t )( C ) : # of customers in virtual buffer with remaining patient time in C ⊂ ( −∞ , ∞ ) Evolution 2 -1 0 -1 4 buffer Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations Measure-valued State Descriptor Virtual buffer R ( t )( C ) : # of customers in virtual buffer with remaining patient time in C ⊂ ( −∞ , ∞ ) Evolution 2 -1 0 -1 4 buffer Richness Q ( t ) = R ( t )(( 0 , ∞ )) R ( t ) = R ( t )(( −∞ , ∞ )) Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations System Dynamics Internal transfer process B ( t ) = E ( t ) − R ( t ) Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations System Dynamics Internal transfer process B ( t ) = E ( t ) − R ( t ) 1 + B ( t ) : index of the next customer to be served Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations System Dynamics Internal transfer process B ( t ) = E ( t ) − R ( t ) 1 + B ( t ) : index of the next customer to be served Stochastic dynamic equations E ( t ) � R ( t )( C ) = δ u i ( C + t − a i ) , C ∈ B ( R ) i = 1 + B ( t ) Z ( t )( C ) = Z ( 0 )( C + t ) B ( t ) C ∈ B ( R + ) � + 1 { u i >τ i − a i } δ v i ( C + t − τ i ) , i = 1 + B ( 0 ) Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations System Dynamics Internal transfer process B ( t ) = E ( t ) − R ( t ) 1 + B ( t ) : index of the next customer to be served δ u i − ( t − a i ) C Stochastic dynamic equations E ( t ) � R ( t )( C ) = δ u i ( C + t − a i ) , C ∈ B ( R ) i = 1 + B ( t ) Z ( t )( C ) = Z ( 0 )( C + t ) B ( t ) C ∈ B ( R + ) � + 1 { u i >τ i − a i } δ v i ( C + t − τ i ) , i = 1 + B ( 0 ) Many-server Queues with Abandonment / Jiheng Zhang

Background Stochastic Model Fluid Model Functional LLN Approximations System Dynamics Internal transfer process B ( t ) = E ( t ) − R ( t ) 1 + B ( t ) : index of the next customer to be served δ u i − ( t − a i ) C Stochastic dynamic equations E ( t ) � R ( t )( C ) = δ u i ( C + t − a i ) , C ∈ B ( R ) i = 1 + B ( t ) Z ( t )( C ) = Z ( 0 )( C + t ) B ( t ) C ∈ B ( R + ) � + 1 { u i >τ i − a i } δ v i ( C + t − τ i ) , i = 1 + B ( 0 ) Many-server Queues with Abandonment / Jiheng Zhang