# Flow analysis

Please read the attachment with the guidelines, word documnt. The formula is attached in Excel

due date 2/16/13 10pm Eastern time

Volunteer Medical Clinic

The Volunteer Medical Clinic is staffed by a Receptionist, a Nurse, and a Physician and provides low acuity medical care on a walk-in basis. Upon arrival and before treatment, the patients must first register with the receptionist. If the receptionist is busy registering another patient, the arriving patient must wait his/her turn. Upon registration, the patient is directed, based on chief complaint, to either the Nurse or the Physician and must wait in a queue if the designated provider is busy. The Nurse or Physician will then provide treatment and the patient will exit the clinic.

In some cases, however, a patient who is initially directed to and treated by the Nurse may then be directed to the Physician for treatment and, in doing so, would join the queue of patients waiting for the Physician. This happens when either the patient truly needs a treatment from both the Nurse and the Physician or in cases where the patient should have been initially directed to the Physician in the first place.

The following data is from a time study and an analysis of patient treatment records:

· Patients arrive according to a random (Poisson) arrival process at the rate of 6 per hour.

· The average time it take for the Receptionist to register a patient is 6 minutes.

· After registration, the fraction of patients who are directed to the Nurse is two-thirds. The other one-third is directed to the Physician.

· Of the patients who are initially directed to and treated by the nurse, 25% of them will then be directed to be seen and treated by the Physician.

· The average treatment time for the Nurse is 10 minutes.

· The average treatment time for the Physician is 16 minutes.

· All service and treatment time distributions are assumed to be exponential (i.e., maximal variation).

Deliverables (a single Word document )

a) Draw a “flow map” of the clinic which illustrates the servers and the directional flows of patients.

b) Compute the rate of arrivals for each of the three servers in the system.

c) Compute the service rates for each of the three servers.

d) Use QueueCalc to estimate the average waiting time for each of the three servers. Note that since arrivals and service times are at maximal variation, you should set both Coefficients of Variation to 1 in the QueueCalc model.

e) Estimate the door-to-door time for the segment of patients who are served by only the Receptionist and the Nurse. Note, the door-to-door time should include waiting times and service times. What portion of the clinic patients are in this segment?

f) Do the same thing you did for part e) for the other two patient segments.

g) What is the weighted average of the door-to-door time for patients of the Volunteer Medical Clinic?

h) From a queuing standpoint, what are all of the possible ways that you could reduce the average door-to-door time?

2

>Instructions Queue

Calculator Dr. Charles Noon cnoonphd@gmail.com A modification of the spreadsheet Queu

e.

XLS by John McClain of Cornell University

## Infinite Queue Appro x imation

Worksheet

This is a Steady State model which means it estimates Long Run averages. Hence, the formulas do not apply for short periods of time. Your inputs always go in the yellow cells, which look like this: Please be careful with your time units. Two of the inputs are rates, and they must have the same time units. For example, suppose the arrival rate is

4

customers per hour, and the average service time is

1 0 minutes

. Then the service rate must also be given in customers per hour, which would be

6

0/

10

3

basic inputs for the infinite queuing model are S, l and m. There are S identical servers, and the queue can hold an unlimited number of customers. The arrival rate of customers is l, and the service rate per server is m. There are two more inputs for this approximation: CV(s) = Coefficient of Variation of Service Times: CV(a) = Coefficient of Variation of Inter-arrival Times (i.e. times between arrivals): Definition: the coefficient of variation is the standard deviation divided by the mean. With the CV(s) = 1.0, the worksheet assumes that the service times are exponentially distribute

d. In many real-world situations, service times have less variation, often as low as CV(s) = 0.1. In some cases processing time doesn’t vary at all, resulting in CV(s) = 0. With CV(a) = 1.0, the worksheet assumes Poisson arrivals. This is equivalent to assuming that the inter-arrival times are exponentially distributed and, by definition, has CV(a) = 1.0. There are many real situations for which this is true, including service calls for equipment failure and demand for emergency services. However, many other cases may have lower relative variability. For example, a final inspector of new cars coming from a paced assembly line would find his/her “customers” (the cars) arriving with almost no variation, so CV(a) would be near zero. Example: City Clinic serves a mix of walk-in and appointment-based patients averaging 4

5

requests per

8

-hour day (5.6

25

per hour). There are two physicians, each capable of serving 25 patients per 8-hour day (3.

12

5 per hour). a. What is the average service time? b. The standard deviation of service time is 0.

16

hours. What is its Coefficient of Variation for service times? c. What is the average inter-arrival time? d.

The standard deviation of inter-arrival time is 0.1 hours. What is its Coefficient of Variation for arrival times? e.

What is the average size of the waiting line, and how long is the average wait? Solution: a.

To serve 25 patients in 8 hours, a physician must average 8/25 = 0.32 hours per patient. b.

CV(s) = Standard Deviation divided by Average = 0.16/0.32 = 0.5 c.

If 45 patients arrive in 8 hours, one arrives every 8/45 = 0.1

7

8 hours. d.

CV(a) = Standard Deviation divided by Average = 0.1/0.

17

8 = 0.562 e.

On the Infinite Queue Approximation worksheet, put in S = 2, l =5.625, m = 3.125, CV(a) = 0.562 and CV(s) = 0.5. This will result in Lq = 2.

18

6 patients waiting, on average, and Wq = 0.3

9

hours waiting, on average.

## Poisson Distribution

Worksheet

For a Poisson Arrivals Process, this worksheet shows the distribution of the number of arrivals that can occur within a time period. The Poisson Arrival Process is characteristic of most walk-in or unscheduled arrival patterns. The only input value is the mean (average) rate of arrivals for a given period of time (can be hours, days, etc…). The chart on the left shows the probabilities associated with the number of arrivals. The chart on the right shows the probabilities associated with the number of arrivals being less than or equal to the given number. For example, if the average rate of patient arrivals to an ER during the overnight shift is 6.3 per hour, then the probability of exactly 9 arrivals is approximately 8% and the probability of there being 5 or fewer arrivals is

40

%.

Infinite Queue Approximation

1

4

5

0

1

1

minutes

0.8

1

 Approximate Formula for Steady-State, Infinite Capacity Queues Basic Inputs: Number of Servers, S = Arrival Rate, l = Average Time Between Arrivals = 0.250 Service Rate Capacity of each server, m = Average Service Time = 0. 20 Coefficient of Variation of Inter-arrival time, CV(a) = Coefficient of Variation of Service time, CV(s) = Basic Outputs: The Waiting Line: Average Number Waiting in Queue (Lq) = 3.200 <== The Approximation Average Waiting Time (Wq) = 0.8 48 Service: Average Utilization of Servers (rho) = 80.00% Average Number of Customers Receiving Service = The Total System (waiting line plus customers being served): Average Number in the System (L) = 4.000 Average Time in System (W) =

Poisson Distribution

4

40 x

0

6389

0 0.0183156389

1

25556

1

44

2

6525

11

2

8103

56

3

3

4 0.1953668148 4

8369352

5

34519

5

0387

6

6

6

7

7

8

8

9

9

10

767

10

11

11

12

12

13

13

14

14

15

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16

16

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18

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21

22

23 0 23 1
24 0 24 1
25 0 25 1
26 0 26 1

0 27 1

28 0 28 1
29

29 1

30

30 1

 ARRIVAL RATE For the input arrival rate, the charts show the probabilities (and cumulative probability) of the number of arrivals within a period. P(x) 0.0183 15 0.073 26 0.091578 19 0. 14 11 0. 23 30 0.1953668148 0.4334701204 0.6 28 0.156 29 0.785 13 0.1041956346 0.8893260 21 0.0595403626 0.9488663842 0.0297701813 0.9786365655 0.0132311917 0.9918677572 0.00529 24 0.9971602339 0.001924537 0.9990847709 0.0006415123 0.9997262832 0.0001973884 0.9999236716 0.0000563967 0.9999800683 0.0000150391 0.9999951074 0.0000037598 0.9999988672 0.0000008847 0.9999997518 0.0000001966 0.9999999484 0.0000000414 0.9999999898 0.0000000083 0.9999999981 0.0000000016 0.9999999997 22 0.0000000003 0.9999999999 27 5.97066907036922E-16 7.96089209382562E-17

Distribution of Arrivals for a period

(Poisson Arrival Process)

P(x) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1.8315638888734179E-2 7.3262555554936715E-2 0.14652511110987343 0.19536681481316456 0.19536681481316456 0.15629345185053165 0.10419563456702111 5.9540362609726345E-2 2.9770181304863173E-2 1.3231191691050298E-2 5.2924766764201195E-3 1.9245369732436798E-3 6.4151232441456E-4 1.9738840751217228E-4 5.6396687860620656E-5 1.5039116762832175E-5 3.7597791907080438E-6 8.8465392722542207E-7 1.9658976160564933E-7 4.1387318232768281E-8 8.2774636465536562E-9 1.5766597422006965E-9 2.8666540767285388E-10 4.9854853508322414E-11 8.3091422513870696E-12 1.3294627602219313E-12 2.0453273234183552E-13 3.0301145532123785E-14 4.3287350760176845E-15 5.9706690703692201E-16 7.9608920938256244E-17

Number of Arrivals

Probability

Cumulative Distribution of Arrivals

P(x) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1.8315638888734179E-2 9.1578194443670893E-2 0.23810330555354431 0.43347012036670884 0.62883693517987338 0.78513038703040505 0.88932602159742613 0.94886638420715252 0.97863656551201572 0.99186775720306597 0.99716023387948605 0.99908477085272973 0.99972628317714429 0.99992367158465645 0.99998006827251706 0.99999510738927988 0.99999886716847064 0.99999975182239786 0.99999994841215945 0.99999998979947768 0.99999999807694129 0.99999999965360098 0.99999999994026634 0.99999999999012124 0.99999999999843037 0.99999999999975986 0.99999999999996436 0.99999999999999467 0.999999999999999 0.99999999999999956 0.99999999999999967 Number of Arrivals

Cumulative Probability