The Erlang-R Queue. Time-Varying QED Queues with Reentrant Customers in Support of Healthcare Staffing

Transcription

1 Motivation Results : 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

2 Motivation 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.

3 Motivation Results The Problem Studied The Problem Studied (08/01/2009) Other Lab Imagine Nurse Physician Administrative reception Vital signs & Anamnesis First Examination Examination A B A A A Labs Imagine: Treatment Treatment Consultation B X-Ray, CT, 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 - Figure 4 - Activities-Resources (Flow) Chart 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? 7

4 Motivation 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?

5 Motivation 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?

6 Motivation Results The Problem Studied Related Work Mandelbaum A., Massey W.A., Reiman M. Strong Approximations for Markovian Service Networks Massey W.A., Whitt W. Networks of Infinite-Server Queues with Nonstationary Poisson Input Green L., Kolesar P.J., Soares J. Improving the SIPP Approach for Staffing Service Systems that have Cyclic Demands Jennings O.B., Mandelbaum A., Massey W.A., Whitt W. Server Staffing to Meet Time-Varying Demand Feldman Z., Mandelbaum A., Massey W.A., Whitt W. Staffing of Time-Varying Queues to Achieve Time-Stable Performance

7 Content Motivation Results Model Definition Staffing Time-Varying Erlang-R Queue (Delay) Model Definition The (Time-Varying) Erlang-R Queue: 2 Arrivals Poiss(λ t) Needy (s t-servers) rate- μ 1 1-p Patient discharge Content (Delay) rate - δ 2 p 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.

8 Motivation Results Model Definition Staffing Time-Varying Erlang-R Queue Patients Arrivals to an Emergency Department Patien nts per hour Hour of day

9 Motivation 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 = λe[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: R(t) = E[ t t S λ(u)du] = E[λ(t S e)]e[s].

10 Motivation 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 = λe[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: R(t) = E[ t t S λ(u)du] = E[λ(t S e)]e[s].

11 Motivation 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 = λe[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: R(t) = E[ t t S λ(u)du] = E[λ(t S e)]e[s].

12 Motivation 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: where, Theorem: R i (t) = E[λ + i (t S i,e )]E[S i ] λ + 1 (t) = λ(t) + E[λ + 2 (t S 2 )] λ + 2 (t) = pe[λ + 1 (t S 1 )] If service times are exponential, R(t) is the solution of the following Fluid ODE: d dt R 1(t) = λ t + δr 2 (t) µr 1 (t), d dt R 2(t) = pµr 1 (t) δr 2 (t).

13 Motivation Results Case Study Analyzing of the offered load function Case Study: Sinusoidal Arrival Rate + λκsin(ωt). Periodic arrival rate: λt = λ 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 Staffingg level λ(t) λ+(t) 20 R1(t) s(t) Time 4.5 5

14 Motivation Results Case Study Analyzing of the offered load function Case Study: Sinusoidal Arrival Rate P(W>0) Simulation of P(Wait) for various β (0.1 β 1.5) 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)

15 Motivation Results Case Study Analyzing of the offered load function Case Study: Sinusoidal Arrival Rate P(W>0) 1 Halfin-Whitt 0.9 Empirical β Relation between P(wait) and β fits steady-state theory!

16 Motivation Results Case Study Analyzing of the offered load function Case Study: Sinusoidal Arrival Rate Simulation results of servers utilization for various β Utilizatio on Time beta 0.1 beta 0.3 beta 0.5 beta 0.7 beta 1.0 beta 1.5 Performance measure is stable! (0.85 Util 0.98)

17 Motivation Results Case Study Analyzing of the offered load function Can We Use Erlang-C? Simulation results of P(wait): Erlang-R vs. Erlang-C and PSA P(W>0) Time Erlang R Erlang C PSA Using Erlang-C s R(t), does not stabilize systems performance.

18 Motivation Results Case Study Analyzing of the offered load function Why Erlang-C Does Not Fit Re-entrant Systems? Compare R(t) of Erlang-C and Erlang-R: Erlang-C offered-load (with concatenated services) : [ ( R(t) = E λ t 1 )] [ ] 1 1 p S 1,e E 1 p S 1 Erlang-R offered-load: [ ( R 1 (t) = E p i λ t S1 i S i 2 1,e) ] S E[S 1 ] Arrivals i=1 Needy 1 1-p p Patient discharge Content 2

19 Motivation Results Case Study Analyzing of the offered load function Comparison between Erlang-C and Erlang-R Erlang-C under- or over-estimates the Erlang-R offered-load d red Load Offer Erlang C Erlang R 65 λ(t) Time Arrival Rate

20 Motivation Results Case Study Analyzing of the offered load function Comparison between Erlang-C and Erlang-R Theorem: The ratio of amplitudes between Erlang-R and Erlang-C is given by (δ 2 + ω 2 )(((1 p)µ) 2 + ω 2 ) ((µ iω)(δ iω) pµδ)((µ + iω)(δ + iω) pµδ) Plot of amplitudes ratio as a function of ω io ude Rati Amplitu Omega

21 Motivation Results Case Study Analyzing of the offered load function Comparison between Erlang-C and Erlang-R Plot of amplitudes ratio as a function of ω io ude Rati Amplitu Omega Erlang-C over-estimate the amplitude of the offered-load. The re-entrant patients stabilize the system. Minimum ratio achieved when: ω = δµ(1 p) (for example ED).

22 Motivation Results Case Study Analyzing of the offered load function Comparison between Erlang-C and Erlang-R Plot of the ratio of phases as a function of ω Phase Ratio Omega Erlang-C under- or over-estimates the time-lag.

23 Motivation Results Case Study Analyzing of the offered load function Comparison between Erlang-C and Erlang-R Erlang-C under- or over-estimates this time-lag depending on the period s length. 105 λ=30, µ=1, δ=0.5, p=2/3, cycles per day= λ=30, µ=1, δ=0.5, p=2/3, cycles per day= d Load Offered Arriva al Rate d Load Offered Arriva al Rate 80 Erlang C Erlang R 26 λ(t) Time 86 Erlang C 26 Erlang R λ(t) Time

24 Motivation Results Case Study Analyzing of the offered load function Small systems - Hospitals Small systems: No of doctors range from 1 to 5 Constrains: Staffing resolution: 1 hour Minimal staffing: 1 doctor per type Integer values: s(t) = [R 1 (t) + β R 1 (t)] Example: R = 2.75 β range s P(W > 0) [0, 0.474] % (0.474, 1.055] % (1.055, 1.658] % (1.658, 2.261] 6 3.0% and up 7 0% => Can not achieve all performance levels!

25 Motivation Results Case Study Analyzing of the offered load function Small systems - Hospitals P(W>0) Time Beta=0.1 Beta=0.5 Beta=1 Beta=1.5 P(Wait) is stable and separable!

26 Motivation Results Case Study Analyzing of the offered load function Conclusions In time-varying systems where patients return for multiple services: 1 Using the MOL (IS) algorithm for staffing stabilizes performance. 2 Re-entrant patients stabilize the system. 3 Using single-service models, such as Erlang-C, is problematic in the re-entrant ED environment: Time-varying arrivals Transient behavior even with constant parameters

27 Motivation Results Case Study Analyzing of the offered load function What next? Fluid and diffusion approximations for mass-casualty events QED - MOL approximations for the processes: Number of customers in system Virtual waiting time Extension: upper limit on the number of customers within the system

28 What next? Motivation Results Content Case Study Analyzing of the offered load function (Delay) The semi-open Closed Erlang-R: queue 2 N beds Arrivals Patient is Needy 1 1-p p Patient is Content Blocked patients 2 Does MOL approximation works? yes, stabilizing performance IW model closed network: is achieved. Is it close to M/M/s/n model? no. Arrivals: exp( λ ),1 Needy: exp( μ ), S

29 Motivation Results Case Study Analyzing of the offered load function Thank You