Flow analysis

Instructions

		Queue Calculator
			Dr. Charles Noon [email protected]
			A modification of the spreadsheet Queue.XLS by John McClain of Cornell University
		Infinite Queue Approximation 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 10 minutes.
		Then the service rate must also be given in customers per hour, which would be 60/10 or 6.
		Your Inputs: The 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 distributed.
		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 45 requests per 8-hour day (5.625 per hour).
		There are two physicians, each capable of serving 25 patients per 8-hour day (3.125 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 = 0.178 hours.
		d.	CV(a) = Standard Deviation divided by Average = 0.1/0.178 = 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 = 2.186 patients waiting, on average, and Wq = 0.39 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

	Approximate Formula for Steady-State, Infinite Capacity Queues
	Basic Inputs:					Number of Servers, S =	1
						Arrival Rate, l =	4	Average Time Between Arrivals =			0.250
						Service Rate Capacity of each server, m =	5	Average Service Time =		0.200
						Coefficient of Variation of Inter-arrival time, CV(a) =	1
						Coefficient of Variation of Service time, CV(s) =	1
	Basic Outputs:
	The Waiting Line:					Average Number Waiting in Queue (Lq) =	3.200	<== The Approximation
						Average Waiting Time (Wq) =	0.8		48	minutes
	Service:					Average Utilization of Servers (rho) =	80.00%
						Average Number of Customers Receiving Service =	0.8
	The Total System (waiting line plus customers being served):
						Average Number in the System (L) =	4.000
						Average Time in System (W) =	1

Poisson Distribution

	ARRIVAL RATE	4		For the input arrival rate, the charts show the probabilities (and cumulative probability) of the number of arrivals within a period.
		40			x	P(x)
					0	0.0183156389	0	0.0183156389
					1	0.0732625556	1	0.0915781944
					2	0.1465251111	2	0.2381033056
					3	0.1953668148	3	0.4334701204
					4	0.1953668148	4	0.6288369352
					5	0.1562934519	5	0.785130387
					6	0.1041956346	6	0.8893260216
					7	0.0595403626	7	0.9488663842
					8	0.0297701813	8	0.9786365655
					9	0.0132311917	9	0.9918677572
					10	0.0052924767	10	0.9971602339
					11	0.001924537	11	0.9990847709
					12	0.0006415123	12	0.9997262832
					13	0.0001973884	13	0.9999236716
					14	0.0000563967	14	0.9999800683
					15	0.0000150391	15	0.9999951074
					16	0.0000037598	16	0.9999988672
					17	0.0000008847	17	0.9999997518
					18	0.0000001966	18	0.9999999484
					19	0.0000000414	19	0.9999999898
					20	0.0000000083	20	0.9999999981
					21	0.0000000016	21	0.9999999997
					22	0.0000000003	22	0.9999999999
					23	0	23	1
					24	0	24	1
					25	0	25	1
					26	0	26	1
					27	0	27	1
					28	0	28	1
					29	5.97066907036922E-16	29	1
					30	7.96089209382562E-17	30	1

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