Question

There is no need to develop a Simio model as part of this homework. You can certainly use a Simio model to validate the results that you obtain using the spreadsheet. For an M/M/1 queue with mean interarrival time 1.25 minutes and mean service time 1 minute, find all five of Wa, W, La, L, and p. For each, interpret in words. Be sure to state all your units (always!), and the relevant time frame of operation. 1. 2. Based on the arrival and service time data in problem 1, suppose that we'd like to see what would happen if the arrival rate were to increase in small steps; maybe a single server barbershop would like to increase business by some advertising or coupons. Create a spreadsheet or a computer program, and re-evaluate all five of Wa, W, La, L, and p, except increasing the arrival rate by 5% over its original value, then by 15% over its original value, then by 20% over its original value, and so on up through increasing it by 100% over its original value (i.e., doubling the arrival rate). Make plots of each of the five metrics as functions of the percent increase in the arrival rate. Discuss. Repeat Problem 1 for an M/M/3 queue with mean interarrival time 1.25 minutes and mean service time 3 minutes at each of the three servers. Hint: You might want to consider creating a computer program or spreadsheet for this problem 3. 4. In Problem 3, increase the arrival rate in the same steps as in Problem 2, and again evaluate all ofWo, W, Lo,L, and p at each step. If you run into any particular problems at some point as you increase the arrival rate, how many more servers (beyond the three that you already have) would you need to calm things down?

Answer 1

1)

l =1.25 mins or 48 customers per hour

u =1 min or 60 customers per hour

Wq =Ws- u = l/(u(u-l)) . = 48/60(12) = 1/15 hours = 4 mins

Ws =Ls/l = =1/(u-l) = 1/(12) =5 mins

Lq =l Wq = l^2/ u(u-l)) = 48*48/ 60 *12 = 16/5 = 3.2

Ls =l/u-l= 48/ (60-48) = 4 patients

rho = l/u =48/60= .8

b)

	in minutes	in rate terms	5% inc	10% inc	15% inc	20% inc	50%inc	100%inc
l	1.25	48	50.4	52.8	55.2	57.6	72	96
u	1	60	60	60	60	60	60	60
		base	5% inc	10% inc	15% inc	20% inc	50%inc	100%inc
	wq (mins)	4	5.25	7.333333333	11.5	24	-6	-2.666666667
	ws (mins)	5	6.25	8.333333333	12.5	25	-5	-1.666666667
	ls	3.2	4.41	6.453333333	10.58	23.04	-7.2	-4.266666667
	lq	4	5.25	7.333333333	11.5	24	-6	-2.666666667
	rho	0.8	0.84	0.88	0.92	0.96	1.2	1.6

3)

1)

l =1.25 mins or 48 customers per hour

u =3 min or 20 customers per hour per server

c =3 servers

po= 1/(1+rho rho^2/2 + rho^c/ c! (1- rho/c)) = 1/(1+.8 + .8^/2 +.8^3/ 6(1- .8/3)) =.058

Wq = lq/l = .0539

Ws = wq +1/u =

Lq = rho^(c+1)/(c-1)! (c-rho)^2) Po = 2.5888

L = lWq + l/u =

rho =l/u =48/20 =2.4

utilization= l/cu =48/3*20= .8

b)

	in minutes	in rate terms	5% inc	10% inc	15% inc	20% inc	50%inc	100%inc
l	1.25	48	50.4	52.8	55.2	57.6	72	96
u	3	20	20	20	20	20	20	20
3		base	5% inc	10% inc	15% inc	20% inc	50%inc	100%inc
	wq (mins)	0.053932584	0.07431789	0.108608731	0.177637726	0.385570844	-0.116546763	-0.063681592
	ws (mins)	0.103932584	0.12431789	0.158608731	0.227637726	0.435570844	-0.066546763	-0.013681592
	ls	4.988764045	6.265621656	8.374541004	12.56560249	25.08888062	-4.791366906	-1.313432836
	lq	2.588764045	3.745621656	5.734541004	9.805602492	22.20888062	-8.391366906	-6.113432836
	rho	2.4	2.52	2.64	2.76	2.88	3.6	4.8
	p0	0.056179775	0.042799058	0.030599755	0.019466615	0.009297136	-0.035971223	-0.074626866
	utilization	0.8	0.84	0.88	0.92	0.96	1.2	1.6

please use the formulas mention in the text I am unable to add spreadsheet to this