1. Chapter 4 Queueing models for capacity planning

2. Plan Introduction to Markov chains Queueing models and key results
Application to hospital capacity planning
A Queuing Approximation Method for Capacity Planning of Emergency Department with Time-Varying Demand

3. N(t) : nb of customers in the queue
Stochastic processes
A stochastic process {Xt, t  T} : a random variable defined on the same state space E and evolving as time t goes on.
Example: the queue length N(t) at time t
Queue
Server
Customer arrival
N(t) : nb of customers in the queue

4. Continuous Time Markov Chain (CTMC)
Stochastic process
Continuous event
Discrete
events
Continuous time
A CTMC is a continuous time and memoryless discrete event stochastic process.
Discrete
time
Memoryless

5. Key conditions for memoryless
Memoryless times
All times (activity times, repair times, lifetimes, …) are exponentially distributed.
X = EXP(m) : P(X  x) = 1 – e-x/m, E[X] = m, Var[X] = m2
Memoryless events
All events (arrivals, machine failures, …) occur according to a POISSON process
A POISSON event e of frequency (also called event rate) l :
time between occurrences of e = EXP(1/ l)

6. A single server queue Queue Exponential service time at rate m
Poisson customer arrival at rate l
N(t) : nb of customers in the queue

7. Markov chain representation
Freq. of event e12 s1
m13
Freq. of event e13
m41
Freq. of event e41 s3 s4

8. A single server queue 1 2 3 Queue Exponential service time at rate m
Poisson customer arrival at rate l
N(t) : nb of customers in the queue l 1 l 2 l 3 l m m m m

9. Steady-state distribution
Steady-state distribution = probability distribution after infinite time
pi = probability of being in state i in steady-state
Alternative definition (under ergodicity condition)
pi = percentage of time of state i over infinite time

10. Determination of the steady-state distribution
p1m12
Probability flow
p1m13
p4m41
Probability flow of event eij
=p1m12
=frequency of event eij
Flow balance equation
Total flow in = Total flow out
Holds for any state or subset of states
Normalisation equation
i pi = 1
p4m41 = p1m12 + p1m13

11. i pi = 1 Online derivation A single server queue 1 2 3l 1 l 2 l 3 l m m m m
Flow balance equation
Total flow in = Total flow out
Normalisation equation
i pi = 1
Online derivation

12. Plan Introduction to Markov chains Queueing models and key results
Application to hospital capacity planning
A Queuing Approximation Method for Capacity Planning of Emergency Department with Time-Varying Demand

13. Definition of a queueing system
Departure of served customers
Customer arrivals
Departure of impatient customers
A queueing system can be described as follows:
"customers arrive for a given service, wait if the service cannot start immediately and leave after being served"
The term "customer" can be men, products, machines, ...

14. Notation of Kendall Kendall notation of queueing systems T/X/C/K/P/Z
– T: probability distribution of inter-arrival times
– X: probability distribution of service times
– C: Number of servers
– K: Queue capacity
– P: Size of the population
– Z: service discipline
In this course,
– K =  : unlimited queue capacity
– P =  : infinity population
– Z = FIFO: First In First Out service

15. Notation of Kendall T/X/C
– T: probability distribution of inter-arrival times
– X: probability distribution of service times
– C: Number of servers
T or X can take the following values:
– M : markovian (i.e. exponential)
– G : general distribution
– D : deterministic
M/M/1 = Markovian arrival & service single server queue
M/M/n = Markovian arrival & service n-servers queue

16. (Number = Throughput  Delay)
Little’s Laws
For any stable system,
L = TH×W
(Number = Throughput  Delay)
where
L : average number of customers in the system
W : average response time
TH : average throughput rate
Queueing system TH TH L W

17. M/M/1 queue Stationary distribution: pn = rn(1-r), n≥0
where r = l/m is called traffic intensity.
Ls = Number of customers in the system = r/(1-r) = l/(m-l)
Ws = Sojourn time in the system = 1/(1-r)m = 1/(m-l)
Lq = queue length = l2/(m-l)m = Ls - r
Wq = average waiting time in the queue = l/(m-l)m = Ws - 1/m
TH = departure rate = l
Server utilization ratio = r
Server idle ratio = P0 = 1 - r
P{n > k} = Probability of more than k customers = rk+1

18. M/M/c queue – Erlang C system
Exponentially distributed service tim l
Poisson arrivals
N(t) : number of customers in the system
N(t) is a birth and death process with
The birth rate l.
The deadth rate is not constant and is equal to N(t)m if N(t)  C and Cm if N(t) > C.
Stability condition : l< cm.

19. M/M/c queue – Erlang C system
Stationary probability distribution:
a= l/m : offered load
= l/cm = a/c: traffic intensity
pn = an/n! p0, " 0 < n  c

20. M/M/c queue – Erlang C system
Ls = Number of customers in the system
= Lq + a
Ws = Sojourn time in the system
= Wq + 1/m
Lq = Average queue length =
Wq = Average waiting time
= Lq / l
= Average number of busy server, p = a

21. M/M/c queue – Erlang C system
C(c,a) = Waiting probability of an incoming customer
= pc + pc
wq = random waiting time of a customer (Moment generating funct)
T = Waiting time target
g(T) = Service level
= P(wq ≤ T)
Erlang C formula

22. M/M/c with impatient customers –Erlang B
Similar to M/M/C queue except the loss of customers which arrive when all servers are busy.
Markov chain of M/M/2 queue with impatient customers

23. M/M/c with impatient customers –Erlang B
Steady state distribution :
a= l/m : offered load
= l/cm : traffic intensity
pn = an/n! p0, " 0 < n  c
Percentage of lost customers = pC
Server utilization ratio = (1 – pC) l/Cm
Insensitivity to service time distribution:
pn depends on the distribution of service time T only through its mean, i.e. with m = E[T]

24. M/M/c with impatient customers –Erlang B
Erlag loss function or Erlang B formula
= Percentage of lost customers or overflow probability
Accepted load

25. M/M/c with impatient customers –Erlang B
Normal approximation for staffing Erlang Loss systems
Condition: high offered load (a > 4) and high targeted service level
N(t) = number of patients : approximately normally distributed
E[N(t)]  a
In M/M/∞ system, N(t) =d POISSON(a), i.e. E[N(t)] = a, Var[N(t)] = a
Square-Root-Staffing-Formula for a delay probability a
Where F is the cdf of the standard normal distribution

26. Computation issues of Erlang B and C formula
!!! recursion for the same offered load !!!

27. Plan Introduction to Markov chains Queueing models and key results
Application to hospital capacity planning
A Queuing Approximation Method for Capacity Planning of Emergency Department with Time-Varying Demand

28. An introductory example
A hospital is exploring the level of staffing needed for a booth in the local mall, where they would test and provide information on the diabetes. Previous experience has shown that, on average, every 6.67 minutes a new person approaches the booth. A nurse can complete testing and answering questions, on average, in twelve minutes.
Assuming s = 2, 3, 4 nurses, a hourly cost of 40€ per nurse and a customer waiting cost of 75€ per hour in the system.
Determine the following: patient arrival rate, service rate, overall system utilisation, nb of patients in the system (Ls), the average queue length (Lq), average time spent in the system (Ws), average waiting time (Wq), probability of no patient, probability of waiting, total system costs. 21

29. An introductory example22

30. Bed capacity of maternity services
Target occupancy level
Consider obsterics units in hospitals. Obsterics is generally operated independently of other services, so its capacity needs can be determined without regard to other services. It is also one for which the use of a standard M/M/s queueing model is quite good. Most obsterics patients are unscheduled and the assumption of Poisson arrivals has been shown to be a ggod one in studies of unscheduled hospital admissions. In addition, the coefficient of variation (CV) of the length of stay (LOS), which is defined as the ratio of the standard deviation to the mean, is typically very close to 1 satisfying the service time assumption of the M/M/s model. 23

31. Bed capacity of maternity services
Since obsterics patients are considered emergent, the American College of Obsterics and Gynecology (ACOG) recommends that occupancy levels of obsterics units not exceeding 75%. Many hospitals have obsterics units operating below this level. However, some have eliminated beds to reduce « excess » capacity and costs and 20% of NY hospitals had obsterics units that would be considered over-utilized by this standard.
Assuming the target occupancy level of 75%, what is the probability of delay for lack of beds for a hospital with s = 10, 20, 40, 60, 80, 100, 150, 200 beds.
Lesson :
For the same occupancy level, the probability of delay decreases with the size of the service. 24

32. Bed capacity of maternity services
Evaluation of capacity based on a delay target leads to very important conclusion. Though there is no standard delay target, it has been suggested that the probability of delay for an obsterics bed should not exceed 1%.
What is the size of an obsterics unit (nb of beds) necessary to achieve a probability of delay not exceeding 1% while keeping the target occupancy level of 60%, 70%, 75%, 80%, 85%?
Lesson :
Achieving high occupancy level while having small probability of delay is only possible for obsterics unit of large hospitals.
Capacity cut should be made with clear understanding of the impact. Simple and naive analysis based on average could lead to bad decisions. 25

33. Bed capacity and seasonality
Impact of seasonality
Consider an obsterics unit with 56 beds which experiences a significant degree of seasonality with occupancy level varying from a low of 68% in January to about 88% in July.
What is the probability of delay in January and in July?
If, as is likely, there are several days when actual arrivals exceed the month average by 10%, what is the probability of delay for these days in July?
Lesson :
Capacity planning should not be based only on the yearly average. Extra bed capacity should be planned for predictable demand increase during peak times. 26

34. Bed capacity reducing through merging
Impact of clinical organisation
Consider the possiblity of combining cardiac and thoracic surgery patients as thoracic patients are relatively few and require similar nursing skills as cardiac patients.
The average arrival rate of cardiac patients is 1,91 bed requests per day and that of thoracic patients is 0,42. No additional information is available on the arrival pattern and we assume Poisson arrivals. The average LOS (Length Of Stay) is 7,7 days for cardiac patients and 3,8 days for thoracic patients.
What is the number of beds for cardiac patients and thoracic patients in order to have average patient waiting time for a bed E(D) not exceeding 0,5, 1, 2, 3 days? What is the number of beds if all patients are treated in the same nursing unit?
Delay in this case measures the time a patient coming out of surgery spends waiting in a recovery unit or ICU until a bed in the nursing unit is available. Long delays cause backups in operating rooms/emergency rooms, surgery cancellation and ambulance diversion. 27

35. Bed capacity reducing through merging
Lesson :
Personal and equipment flexibility and service pooling can achieve higher occupancy level and reduction of beds.
However, priority given to one patient group could significantly degrade the waiting time of other patients if all treated in the same nursing unit. 27

36. Staffing Emergency Department under Service Level Constraints
You are asked to help improving the nurse planning of an Emergency Department (ED). From the historic data, you are able to obtain the following demand forecast on the number arrivals at the ED:
From the statistics, 60% of the ED patients are regular patients and need only 15 minutes of the nursing care. However 40% of the ED patients are true emergency patients and require about 1h nursing care at ED before transfer to the wards. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0,5 27

37. Staffing Emergency Department under Service Level Constraints
Patient arrivals 27

38. Staffing Emergency Department under Service Level Constraints
Preliminary goals :
Derive the workload profile of a typical day.
Enumerate all possible shift patterns. Shifts of 8h start either 7h-9h (20€/h), or 15h-17h (22€/h), or 23-01h (25€/h). Shifts of 12h start either 7h-9h (21€/h) or 19h-21h (23€/h).
Goal 1: Planning shifts to meet loss probability target (<5%, 1%) (Erlang B)
Goal 2: Planning shifts to meet waiting time targets (Erlang C)
1: less than 20 minutes for at least 80% of patients
2: less than 1h for at least 95% of patients

39. Plan
Introduction to Markov chains
Queueing models and key results
Application to hospital capacity planning
A Queuing Approximation Method for Capacity Planning of Emergency Department with Time-Varying Demand

40. Qiang LIU, Xiaolan XIE, Ran LIU
A Queuing Approximation Method for Capacity Planning of Emergency Department with Time-Varying Demand
Qiang LIU, Xiaolan XIE, Ran LIU

41. Outline Background Problem description and solution Evaluating model
Scheduling model
Conclusions and future work

42. Arrival rates from the emergency department in Ruijin hospital
Background
Emergency department has to provide timely medical service 24 hours round .
At least one physician in ED at anytime.
Patient arrival rate fluctuates significantly during a day
Arrival rates from the emergency department in Ruijin hospital

43. Background
Current capacity plan does not take into account fluctuation of patient arrival.
Crowding during peak hours;
Low human resource utilization during low arrival periods;
A more efficient plan could be made if we set the starting working time and shift length flexible.

44. Outline Background Problem description and solution Evaluating model
Scheduling model
Conclusions and future work

45. Problem description and solution
Reduce patient waiting time
Relief the crowding in ED
Improve staff utilization
Time-varying patient arrival rate
Flexible working time and shift length of physicians.
Mathematical Modeling
Capacity Schedule
Key Point:
Since the patient arrival rate fluctuates and the number of doctors is also time-dependent, the system cannot reach any steady state.
Results of classic queuing theory do not apply directly to evaluate patient waiting time.

46. Outline Background Problem description and solution Evaluation model
Scheduling model
Conclusions and future work

47. Evaluation model △t Length of each period
λt # of patients arrived in period,
ut # of patients served in period
pt # of physicians in period
qt # of patients overflowed
Flow balance equation qt = qt-1 + lt - ut
Waiting time of served patients
W(ut, pt) = steady-state waiting time of M/M/C model with arrival rate ut and pt servers.
Similar to existing Pointwise Stationary Approximation (PSA)
Waiting time of overflowed patients Dt

48. Solved by linearization of the non-linear function.
Evaluation model …. ….
Non-linear
# of patients served not exceeding physician service rate
The given schedule
Solved by linearization of the non-linear function.

49. Result of evaluation model
The current capacity plan is to be evaluated
Number of patients served in each period
Waiting time in each period

50. outline Background Problem description and solution Evaluating model
Scheduling model
Conclusions and future work

51. Scheduling model Shift decision variables:
st = number of physicians starting their shift in period t
et = number of physicians completing their shift in period t

52. Scheduling model Evaluation model All shifts start and finish in a day
# of physicians and shifts
Shift length  [L, U]
Shifts will overlap each other
N physicians in total

53. Numerical Experiments
N =10 physicians
Shift length [5h, 8h]
Service rate m =
Hourly periods
Arrival rate collected from hospital history data

54. Numerical Experiments
New vs actual schedules
# of patients served and delayed
Actual schedule
# of patients served and delayed
New schedule

55. outline Background Problem description and solution Evaluating model
Scheduling model
Conclusions and future work

56. Conclusions
An evaluation model for a given shift schedule that combines steady-state performance and controlled overflow.
A shift optimization model based on the evaluation model.
Resulting staffing schedule better matches the fluctuating patient arrival.
Future research
Improve the accuracy of the evaluation model, especially when the congestion level is not very high.
Design fast solution techniques for solving the evaluation and shift optimization models.