USING SIMULATION TO STUDY SERVICE-RATE CONTROLS TO STABILIZE - PowerPoint PPT Presentation

USING SIMULATION TO STUDY SERVICE-RATE CONTROLS TO STABILIZE PERFORMANCE IN A SINGLE-SERVER QUEUE WITH TIME-VARYING ARRIVAL RATE Ni Ma and Ward Whitt Columbia University December 5, 2015 Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 1 / 31

Outline Motivation 1 Stabilizing Performance Service-Rate Controls The Model 2 Simulation Methods For Nonstationary Models 3 Generating the Arrival Process Generating the Service Times Simulation Experiments 4 Simulation Results 5 Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 2 / 31

Motivation Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 3 / 31

Motivation: Stabilizing Performance Given the time-varying arrival rates, we are interested in an algorithm that can stabilize performance of the queueing system, e.g. expected delay, delay probability, expected queue length. Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 4 / 31

Motivation: Stabilizing Performance Given the time-varying arrival rates, we are interested in an algorithm that can stabilize performance of the queueing system, e.g. expected delay, delay probability, expected queue length. Earlier papers that study server-staffing to stabilize performance in multi-server queues with time-varying arrivals. Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 4 / 31

Motivation: Stabilizing Performance Given the time-varying arrival rates, we are interested in an algorithm that can stabilize performance of the queueing system, e.g. expected delay, delay probability, expected queue length. Earlier papers that study server-staffing to stabilize performance in multi-server queues with time-varying arrivals. M. Defraeye and I. Van Niewenhuyse (2013) Controlling excessive waiting times in small service systems with time-varying demand: an extension of the ISA algorithm. Decision Support Systems 54(4), 1558 – 1567. Y. Liu and W. Whitt (2012) Stabilizing customer abandonment in many-server queues with time-varying arrivals. Oper. Res. 60(6), 1551 – 1564. O.B. Jennings, A. Mandelbaum, W.A. Massey and W. Whitt (1996) Server staffing to meet time-varying demand. Manag. Sci. 42(10), 1383 –1394. Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 4 / 31

Motivation: Service-Rate Controls Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 5 / 31

Motivation: Service-Rate Controls Problem : systems with only a few servers or with inflexible staffing. Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 5 / 31

Motivation: Service-Rate Controls Problem : systems with only a few servers or with inflexible staffing. In many applications, it is possible to change the processing rate . Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 5 / 31

Motivation: Service-Rate Controls Problem : systems with only a few servers or with inflexible staffing. In many applications, it is possible to change the processing rate . Example (use a service-rate control) 1 Hospital Surgery Rooms 2 Airport Security Inspection Lines Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 5 / 31

Our Contributions Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 6 / 31

Our Contributions We use simulation to study service-rate controls to stabilize performance in a single-server queue with time-varying arrival rates. Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 6 / 31

Our Contributions We use simulation to study service-rate controls to stabilize performance in a single-server queue with time-varying arrival rates. We conduct simulation experiments to evaluate the performance of alternative service-rate controls. Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 6 / 31

Our Contributions We use simulation to study service-rate controls to stabilize performance in a single-server queue with time-varying arrival rates. We conduct simulation experiments to evaluate the performance of alternative service-rate controls. We develop an efficient algorithm for simulating a time-varying queue with a service-rate control. Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 6 / 31

The Model Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 7 / 31

The G t / G t / 1 queue G t / G t / 1 Single-Server Queueing Model Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 8 / 31

The G t / G t / 1 queue G t / G t / 1 Single-Server Queueing Model Single server Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 8 / 31

The G t / G t / 1 queue G t / G t / 1 Single-Server Queueing Model Single server Time-varying arrival rate function Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 8 / 31

The G t / G t / 1 queue G t / G t / 1 Single-Server Queueing Model Single server Time-varying arrival rate function First-Come First-Served service policy Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 8 / 31

The G t / G t / 1 queue G t / G t / 1 Single-Server Queueing Model Single server Time-varying arrival rate function First-Come First-Served service policy Unlimited waiting space Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 8 / 31

The G t / G t / 1 queue G t / G t / 1 Single-Server Queueing Model Single server Time-varying arrival rate function First-Come First-Served service policy Unlimited waiting space Service rate is subject to control Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 8 / 31

The G t / G t / 1 queue G t / G t / 1 Single-Server Queueing Model Single server Time-varying arrival rate function First-Come First-Served service policy Unlimited waiting space Service rate is subject to control i.i.d. service requirements separate from the service rate Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 8 / 31

The Arrival Process Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 9 / 31

The Arrival Process A time-transformation of a stationary counting process: � t A ( t ) ≡ N a (Λ( t )) ≡ N a ( λ ( s ) ds ) , t ≥ 0 , (1) 0 where Λ is the cumulative arrival rate function: � t Λ( t ) = 0 λ ( s ) ds , t ≥ 0. Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 9 / 31

The Arrival Process A time-transformation of a stationary counting process: � t A ( t ) ≡ N a (Λ( t )) ≡ N a ( λ ( s ) ds ) , t ≥ 0 , (1) 0 where Λ is the cumulative arrival rate function: � t Λ( t ) = 0 λ ( s ) ds , t ≥ 0. N a is a rate-1 counting process with unit jumps. Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 9 / 31

The Arrival Process A time-transformation of a stationary counting process: � t A ( t ) ≡ N a (Λ( t )) ≡ N a ( λ ( s ) ds ) , t ≥ 0 , (1) 0 where Λ is the cumulative arrival rate function: � t Λ( t ) = 0 λ ( s ) ds , t ≥ 0. N a is a rate-1 counting process with unit jumps. � t check: E [ A ( t )] = E [ N a (Λ( t ))] = Λ( t ) = 0 λ ( s ) ds . Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 9 / 31

The Arrival Process A time-transformation of a stationary counting process: � t A ( t ) ≡ N a (Λ( t )) ≡ N a ( λ ( s ) ds ) , t ≥ 0 , (1) 0 where Λ is the cumulative arrival rate function: � t Λ( t ) = 0 λ ( s ) ds , t ≥ 0. N a is a rate-1 counting process with unit jumps. � t check: E [ A ( t )] = E [ N a (Λ( t ))] = Λ( t ) = 0 λ ( s ) ds . All the stochastic variability is separated from the deterministic arrival rate function. Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 9 / 31

The Service Process Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 10 / 31

The Service Process Queue Length and Departure Process Q ( t ) ≡ A ( t ) − D ( t ) , t ≥ 0 , (2) � t D ( t ) ≡ N s ( µ ( s )1 { Q ( s ) > 0 } ds ) , t ≥ 0 , (3) 0 Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 10 / 31

The Service Process Queue Length and Departure Process Q ( t ) ≡ A ( t ) − D ( t ) , t ≥ 0 , (2) � t D ( t ) ≡ N s ( µ ( s )1 { Q ( s ) > 0 } ds ) , t ≥ 0 , (3) 0 where Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 10 / 31

The Service Process Queue Length and Departure Process Q ( t ) ≡ A ( t ) − D ( t ) , t ≥ 0 , (2) � t D ( t ) ≡ N s ( µ ( s )1 { Q ( s ) > 0 } ds ) , t ≥ 0 , (3) 0 where N s is a rate-1 counting process with unit jumps, independent of N a . Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 10 / 31

The Service Process Queue Length and Departure Process Q ( t ) ≡ A ( t ) − D ( t ) , t ≥ 0 , (2) � t D ( t ) ≡ N s ( µ ( s )1 { Q ( s ) > 0 } ds ) , t ≥ 0 , (3) 0 where N s is a rate-1 counting process with unit jumps, independent of N a . � t E [ D ( t ) | Q ( s ) , 0 ≤ s ≤ t ] = 0 µ ( s )1 { Q ( s ) > 0 } ds . Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 10 / 31