1. Queues in Hospitals: Semi-Open Queueing Networks in the QED Regime Galit Yom-Tov Joint work with Avishai Mandelbaum 31/Dec/2008 Technion – Israel Institute of Technology

2. 2 Agenda Introduction Introduction Medical Unit Model Medical Unit Model Mathematical Results Mathematical Results Numerical Example Numerical Example Time-varying Model Time-varying Model Future Research Future Research

3. 3 Work-Force and Bed Capacity Planning Total health expenditure as percentage of gross domestic product: Israel 8%, EU 10%, USA 14%. Total health expenditure as percentage of gross domestic product: Israel 8%, EU 10%, USA 14%. Human resource constitute 70% of hospital expenditure. Human resource constitute 70% of hospital expenditure. There are 3M registered nurses in the U.S. but still a chronic shortage. There are 3M registered nurses in the U.S. but still a chronic shortage. California law set nurse-to-patient ratios such as 1:6 for pediatric care unit. California law set nurse-to-patient ratios such as 1:6 for pediatric care unit. O.B. Jennings and F. de Véricourt (2008) showed that fixed ratios do not account for economies of scale. O.B. Jennings and F. de Véricourt (2008) showed that fixed ratios do not account for economies of scale. Management measures average occupancy levels, while arrivals have seasonal patterns and stochastic variability (Green 2004). Management measures average occupancy levels, while arrivals have seasonal patterns and stochastic variability (Green 2004).

4. 4 Research Objectives Analyzing model for a Medical Unit with s nurses and n beds, which are partly/fully occupied by patients: semi-open queueing network with multiple statistically identical customers and servers. Analyzing model for a Medical Unit with s nurses and n beds, which are partly/fully occupied by patients: semi-open queueing network with multiple statistically identical customers and servers. Questions addressed: How many servers (nurses) are required (staffing), and how many fixed resources (beds) are needed (allocation) in order to minimize costs while sustaining a certain service level? Questions addressed: How many servers (nurses) are required (staffing), and how many fixed resources (beds) are needed (allocation) in order to minimize costs while sustaining a certain service level? Coping with time-variability Coping with time-variability

5. 5 We Follow - Basic: Basic: Halfin and Whitt (1981) Halfin and Whitt (1981) Mandelbaum, Massey and Reiman (1998) Mandelbaum, Massey and Reiman (1998) Khudyakov (2006) Analytical models in HC: Analytical models in HC: Nurse staffing: Jennings and Véricourt (2007), Yankovic and Green (2007) Nurse staffing: Jennings and Véricourt (2007), Yankovic and Green (2007) Beds capacity: Green (2002,2004) Beds capacity: Green (2002,2004) Service Engineering (mainly call centers): Service Engineering (mainly call centers): Gans, Koole, Mandelbaum: “Telephone call centers: Tutorial, Review and Research prospects” Gans, Koole, Mandelbaum: “Telephone call centers: Tutorial, Review and Research prospects”

6. 6 The MU Model as a Semi-Open Queueing Network Patient is Dormant 1-p p 1 3 2 Arrivals from the ED ~ Poiss(λ) Blocked patients Patient is Needy Patient discharged, Bed in Cleaning N beds Service times are Exponential; Routing is Markovian

7. 7 The MU Model as a Closed Jackson Network p Dormant: ·\M\∞, δ 1-p 3 1 2 4 Cleaning: ·\M\∞, γ Arrivals: ·\M\1, λ Needy: ·\M\s, µ N →Product Form - π N (i,j,k) stationary dist.

8. 8 Stationary Distribution

9. 9 Service Level Objectives ( Function of λ,μ,δ,γ,p,s,n) Blocking probability Blocking probability Delay probability Delay probability Probability of timely service (wait more than t) Probability of timely service (wait more than t) Expected waiting time Expected waiting time Average occupancy level of beds Average occupancy level of beds Average utilization level of nurses Average utilization level of nurses

10. 10 Blocking probability The probability to have l occupied beds in the ward: The probability to have l occupied beds in the ward:

11. 11 Probability of timely service and the Delay Probability What happens when a patient becomes needy ? He will need to wait an in-queue random waiting time that follows an Erlang distribution with (i-s+1) + stages, each with rate μ s. What is the probability that this patient will find i other needy patients? We need to use the Arrival Theorem

12. 12 1 Needy: ·\M\s, µ Patient become Needy s nurse s i Needy patients p Dormant: ·\M\∞, δ 1-p 3 2 4 Cleaning: ·\M\∞, γ Arrivals: ·\M\1, λ N

13. 13 The Arrival Theorem Thus, the probability that a patient that become needy will see i needy patients in the system is π n-1 (i,j,k)

14. 14 Probability of timely service and the Delay Probability

15. 15 Expected waiting time via the tail formula:

16. 16 QED Q’s: Quality- and Efficiency-Driven Queues Traditional queueing theory predicts that service- quality and server’s efficiency must trade off against each other. Traditional queueing theory predicts that service- quality and server’s efficiency must trade off against each other. Yet, one can balance both requirements carefully (Example: in well-run call-centers, 50% served “immediately”, along with over 90% agent’s utilization, is not uncommon) Yet, one can balance both requirements carefully (Example: in well-run call-centers, 50% served “immediately”, along with over 90% agent’s utilization, is not uncommon) This is achieved in a special asymptotic operational regime – the QED regime This is achieved in a special asymptotic operational regime – the QED regime

17. 17 QED Regime characteristics High service quality High service quality High resource efficiency High resource efficiency Square-root staffing rule Square-root staffing rule Many-server asymptotic Many-server asymptotic The offered load at service station 1 (needy) The offered load at non- service station 2+3 (dormant + cleaning)

18. 18 The probability is a function of three parameters: beta, eta, and offered-load-ratio The probability is a function of three parameters: beta, eta, and offered-load-ratio Probability of Delay

19. 19 Expected Waiting Time Waiting time is one order of magnitude less then the service time. Waiting time is one order of magnitude less then the service time.

20. 20 Probability of Blocking P(Blocking) << P(Waiting) P(Blocking) << P(Waiting)

21. 21 Approximation vs. Exact Calculation – Medium system (n=35,50), P(W>0)

22. 22 Approximation vs. Exact Calculation – P(blocking) and E[W]

23. 23 The influence of β and η? Blocking Waiting eta:

24. 24 = 0.5 = 0.33

25. 25 Numerical Example (based on Lundgren and Segesten 2001 + Yankovic and Green 2007 ) N=42 with 78% occupancy N=42 with 78% occupancy ALOS = 4.3 days ALOS = 4.3 days Average service time = 15 min Average service time = 15 min 0.4 requests per hour 0.4 requests per hour => λ = 0.32, μ=4, δ=0.4, γ=4, p=0.975 => λ = 0.32, μ=4, δ=0.4, γ=4, p=0.975 => Ratio of offered load = 0.1 => Ratio of offered load = 0.1

26. 26 How to find the required β and η? if β=0.5 and η=0.5 (s=4, n=38): P(block)  0.07, P(wait)  0.4 if β=1.5 and η  0 (s=6, n=37): P(block)  0.068, P(wait)  0.084 if β=-0.1 and η  0 (s=3, n=34): P(block)  0.21, P(wait)  0.70

27. 27 E[W]P(blocking)P(wait)SNRegime 0.110.840.3247Nurses – QED; Beds-ED 00.83077Nurses – QD ; Beds-ED 0.110.550.441020Nurses – QED; Beds-QED 00.5301520Nurses – QD ; Beds-QED 7.890.851330Nurses – ED ; Beds-ED 0.130.350.551430Nurses – QED; Beds-QED 00.3102130Nurses – QD ; Beds-QED 14.560.851350Nurses – ED ; Beds-ED 2.850.511050Nurses – ED ; Beds-QED 0.120.050.52150Nurses – QED; Beds-QD 00.0203150Nurses – QD ; Beds-QD Lambda-10; Delta-0.5; Mu-1; Gamma-10; p-0.5; Ratio=1.05

28. 28 Modeling time-variability Procedures at mass-casualty event Procedures at mass-casualty event Blocking cancelled -> open system Blocking cancelled -> open system Patient is Dormant 1-p p 1 2 Arrivals from/to the EW ~ Poiss(λ t ) Patient is Needy

29. 29 Fluid and Diffusion limits Patient is Dormant 1-p p 1 2 Arrivals from/to the EW ~ Poiss(λ t ) Patient is Needy

30. 30 Scaling The arrival rate and the number of nurses grow together to infinity, i.e. scaled up by η.

31. 31 Fluid limits

32. 32 Special case – Fixed parameters The differential equations become: The differential equations become: Steady-state analysis: What happens when t→∞? Steady-state analysis: What happens when t→∞?

33. 33 Steady-state analysis Overloaded - Underloaded - Overloaded - Underloaded -

34. 34 Critically Loaded System: Area 1 (q1<ŝ) Area 2 (q1≥ŝ)

35. 35 Steady-state analysis - Summery

36. 36 Diffusion limits

37. 37 Diffusion limits Using the ODE one can find equations for: Using the ODE one can find equations for: And to use them in analyzing time-variable system. For example: And to use them in analyzing time-variable system. For example:

38. 38

39. 39 Future Research Investigating approximation of closed system Investigating approximation of closed system From which n are the approximations accurate? (simulation vs. rates of convergence) From which n are the approximations accurate? (simulation vs. rates of convergence) Optimization Optimization Solving the bed-nurse optimization problem Solving the bed-nurse optimization problem Difference between hierarchical and simultaneous planning methods Difference between hierarchical and simultaneous planning methods Validation of model using RFID data Validation of model using RFID data Expanding the model (Heterogeneous patients; adding doctors) Expanding the model (Heterogeneous patients; adding doctors)

40. Thank you

41. 41 Mandating minimum nurse-to-patient ratios Advantage: Ensures patient safety, reduces patients deaths, recruitment of new nurses, and stopping attrition. Disadvantage: Ratios are not flexible enough to allow hospitals to operate safely and efficiently The Wall Street Journal - Dec 2006

42. 42 An Optimization Problem