Motivation The Erlang-R Queue Results The Erlang-R Queue: Time-Varying QED Queues with Reentrant Customers in Support of Healthcare Staffing Galit Yom-Tov Avishai Mandelbaum Industrial Engineering and Management Technion MSOM Conference, June 2010 Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue
Motivation The Erlang-R Queue Results The Problem Studied The Problem Studied Problems in Emergency Departments: Hospitals do not manage patients flow. Long waiting times in the ED for physicians, nurses, and tests. => Deterioration in medical state. Patients leave ED without being seen or abandon during their stay. => Patient return in severe state. We use Service Engineering approach to reduce these effects. Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue
Motivation The Erlang-R Queue Results The Problem Studied The Problem Studied Other Lab Imagine Nurse Physician First First Administrative Vital signs & Examination reception Anamnesis Examination A A A A B Treatment Treatment Labs Imagine: B X-Ray, CT, Consultation Ultrasound B C Follow-up C Decision C Awaiting evacuation Instruction prior discharge Administrative release Alternative Operation - C Recourse Queue - Synchronization Queue – Ending point of alternative operation - Can we determine the number of physicians (and nurses) needed to improve patients flow, and control the system in balance between service quality and efficiency? Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue
μ μ = μ ∧ λ λ = λ δ δ = δ ∞ Motivation The Erlang-R Queue Results The Problem Studied The Problem Studied Standard assumption in service models: service time is continuous. But we find systems in which: service is dis-continuous and customers re-enter service again and again. Service (Needy) Content What is the appropriate staffing procedure? What is the significance of the re-entering customers? What is the implication of using simple Erlang-C models for staffing? Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue
μ μ = μ ∧ λ λ = λ δ δ = δ ∞ Motivation The Erlang-R Queue Results The Problem Studied The Problem Studied Standard assumption in service models: service time is continuous. But we find systems in which: service is dis-continuous and customers re-enter service again and again. Service (Needy) Content What is the appropriate staffing procedure? What is the significance of the re-entering customers? What is the implication of using simple Erlang-C models for staffing? Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue
Motivation The Erlang-R Queue Results The Problem Studied Related Work Mandelbaum A., Massey W.A., Reiman M. Strong Approximations for Markovian Service Networks . 1998. Massey W.A., Whitt W. Networks of Infinite-Server Queues with Nonstationary Poisson Input . 1993. Green L., Kolesar P .J., Soares J. Improving the SIPP Approach for Staffing Service Systems that have Cyclic Demands . 2001. Jennings O.B., Mandelbaum A., Massey W.A., Whitt W. Server Staffing to Meet Time-Varying Demand . 1996. Feldman Z., Mandelbaum A., Massey W.A., Whitt W. Staffing of Time-Varying Queues to Achieve Time-Stable Performance . 2007. Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue
Motivation The Erlang-R Queue Results Model Definition Staffing Time-Varying Erlang-R Queue Model Definition The (Time-Varying) Erlang-R Queue: Needy (s t -servers) rate- μ Arrivals 1-p 1 Patient discharge Poiss( λ t ) p Content (Delay) rate - δ 2 λ t - Arrival rate of a time-varying Poisson arrival process. µ - Service rate. δ - Delay rate (1 /δ is the delay time between services). p - Probability of return to service. s t - Number of servers at time t . Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue
Motivation The Erlang-R Queue Results Model Definition Staffing Time-Varying Erlang-R Queue Patients Arrivals to an Emergency Department 2004-2008 18 16 r hour 14 12 nts per 10 8 Patien 6 4 2 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Hour of day Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue
Motivation The Erlang-R Queue Results Model Definition Staffing Time-Varying Erlang-R Queue Staffing: Determine s t , t ≥ 0 Based on the QED-staffing formula: √ s = R + β R , where R = λ E [ S ] � In time-varying environments: s ( t ) = R ( t ) + β R ( t ) , where β is chosen according to the steady-state QED . Two approaches to calculate the time-varying offered load ( R ( t ) ): PSA / SIPP (lag-SIPP) - divide the time-horizon to planning intervals, calculate average arrival rate and steady-state offered-load for each interval, then staff according to steady-state recommendation (i.e., R ( t ) ≈ ¯ λ ( t ) E [ S ] ). MOL/IS - assuming no constraints on number of servers, calculate the time-varying offered-load. For example, in a single service system: � t R ( t ) = E [ t − S λ ( u ) du ] = E [ λ ( t − S e )] E [ S ] . Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue
Motivation The Erlang-R Queue Results Model Definition Staffing Time-Varying Erlang-R Queue Staffing: Determine s t , t ≥ 0 Based on the QED-staffing formula: √ s = R + β R , where R = λ E [ S ] � In time-varying environments: s ( t ) = R ( t ) + β R ( t ) , where β is chosen according to the steady-state QED . Two approaches to calculate the time-varying offered load ( R ( t ) ): PSA / SIPP (lag-SIPP) - divide the time-horizon to planning intervals, calculate average arrival rate and steady-state offered-load for each interval, then staff according to steady-state recommendation (i.e., R ( t ) ≈ ¯ λ ( t ) E [ S ] ). MOL/IS - assuming no constraints on number of servers, calculate the time-varying offered-load. For example, in a single service system: � t R ( t ) = E [ t − S λ ( u ) du ] = E [ λ ( t − S e )] E [ S ] . Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue
Motivation The Erlang-R Queue Results Model Definition Staffing Time-Varying Erlang-R Queue Staffing: Determine s t , t ≥ 0 Based on the QED-staffing formula: √ s = R + β R , where R = λ E [ S ] � In time-varying environments: s ( t ) = R ( t ) + β R ( t ) , where β is chosen according to the steady-state QED . Two approaches to calculate the time-varying offered load ( R ( t ) ): PSA / SIPP (lag-SIPP) - divide the time-horizon to planning intervals, calculate average arrival rate and steady-state offered-load for each interval, then staff according to steady-state recommendation (i.e., R ( t ) ≈ ¯ λ ( t ) E [ S ] ). MOL/IS - assuming no constraints on number of servers, calculate the time-varying offered-load. For example, in a single service system: � t R ( t ) = E [ t − S λ ( u ) du ] = E [ λ ( t − S e )] E [ S ] . Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue
Motivation The Erlang-R Queue Results Model Definition Staffing Time-Varying Erlang-R Queue The Offered-Load Offered-Load in Erlang-R = The number of busy servers (or the number of customers) in a corresponding ( M t / M / ∞ ) 2 network. Theorem: (Massey and Whitt 1993) R ( t ) = ( R 1 ( t ) , R 2 ( t )) is determined by the following expression: R i ( t ) = E [ λ + i ( t − S i , e )] E [ S i ] where, λ + 1 ( t ) = λ ( t ) + E [ λ + 2 ( t − S 2 )] λ + 2 ( t ) = pE [ λ + 1 ( t − S 1 )] Theorem: If service times are exponential, R ( t ) is the solution of the following Fluid ODE: d dt R 1 ( t ) = λ t + δ R 2 ( t ) − µ R 1 ( t ) , d dt R 2 ( t ) = p µ R 1 ( t ) − δ R 2 ( t ) . Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue
Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function Case Study: Sinusoidal Arrival Rate Periodic arrival rate: λ t = ¯ λ + ¯ λκ sin ( ω t ) . ¯ λ is the average arrival rate, κ is the relative amplitude, and ω is the frequency. External / Internal arrivals rate, Offered-load, and Staffing 120 100 80 g level Staffing 60 60 40 40 λ (t) λ +(t) 20 20 R1(t) s(t) 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue
Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function Case Study: Sinusoidal Arrival Rate Simulation of P(Wait) for various β ( 0 . 1 ≤ β ≤ 1 . 5 ) 1 0.9 0.8 0.7 0.6 P(W>0) 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100105110115 Time [Hour] beta=0.1 beta=0.3 beta=0.5 beta=0.7 beta=1 beta=1.5 Performance measure is stable! ( 0 . 15 ≤ P ( Wait ) ≤ 0 . 85 ) Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue
Motivation The Erlang-R Queue Results Case Study Analyzing of the offered load function Case Study: Sinusoidal Arrival Rate 1 Halfin-Whitt 0.9 Empirical 0.8 0.7 0.6 P(W>0) 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 β Relation between P(wait) and β fits steady-state theory! Galit Yom-Tov, Avishai Mandelbaum The Erlang-R Queue
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