CSCI 8710
Homework #5

Solution

All problems are in "Performance by Design" by Menasce, Almeida, and Dowdy

Ch. 6, exercises 1-4
Ch. 7, exercises 3-4
Ch. 8, exercises 1,8
Ch. 10, exercises 1,6,11

Ch. 6, exercise 1 Using the data in the file WSData.xls , compute the basic statistics for the file size for the set of all files downloaded (PDF and ZIP) taken together. Compute the coefficient of variation of the file size. Compare the results obtained with the separate statistics for PDF and ZIP files shown in this chapter.

Basic statistics:
	Mean:			    835.83
	Median:		           1048
	Standard Deviation:         389.58
	Sample variance:	151,773.70
	Maximum:		  1,299.88 
	Minimum:	            300.44
	Range:	                    999.44
	Sum:		        835,833
	Count:	                   1000
	1/2 95% Conf interval:       24.15
 	Coefficient of variation:     0.47	 


For the two-class model, the COV for the PDF files is only 0.114, and for the
ZIP files it is even smaller, 0.074.  This means that the combined model is
a much looser representation of the actual workload than are the individual
models of the PDF and ZIP file classes.

6-2 Use the results in problem 1 and construct a single class model. Calculate new service demands. Solve the new model and plot its throughput and average download times. Assess the error made by assuming this single class model when the actual system is multiclass. What is an appropriate SLA for this single class model?


To calculate the new model, we might form a weighted average of the service demands
of the two classes (weighted by number of points)  (or just use the initial 
measurements)... to get:

D_cpu =  ( (0.0394 * 411)  + (0.1280 * 589)) / 1000 =  0.087
D_d1,d2 =  ( (0.0771 * 411)  + (0 * 589)) / 1000 = 0.032
D_d3,d4  =  ( (0 * 411)  + (0.2358 * 589)) / 1000 =  0.139

... and then solve using ClosedQN and plot ...

	
	N		X           R		 max_device_utilization
---------------------------------------------------------------------
	2		3.67       0.54476	d3 and d4 at 0.51
	4		5.06	   0.79081	d3 and d4 at 0.70308
	5		5.44       0.91863	d3 and d4 at 0.75656			
	10              6.32	   1.58131      d3 and d4 at 0.87902
	20	        6.78	   2.94796      d3 and d4 at 0.94303
	30		6.93	   4.32890	d3 and d4 at 0.96329
	40		7.00	   5.71413	d3 and d4 at 0.97303

... approaching 100% utilization as we get close to 180 - 200



Or better yet, form a weighted average of the service demands
of the two classes (weighted by number of concurrent jobs)  to get:

D_cpu =  ( (0.0394 * 3.7)  + (0.1280 * 16.1)) / 19.8 =  0.105
D_d1,d2 =  ( (0.0771 * 3.7)  + (0 * 16.1)) / 19.8 = 0.014
D_d3,d4  =  ( (0 * 3.7)  + (0.2358 * 16.1)) / 19.8 =  0.192

... and then solve using ClosedQN and plot ...

in either case we find that the estimates of the bottleneck 
and max capacity are misleading ...

6-3 Use the data in the WSData.xls workbook and compute the 25th percentile and 7th percentile of the PDF and ZIP files. Draw a box and whisker plot for each data set.


See:  hw5figs.pdf 




 6-4  Compute a 90% confidence interval for the size of PDF and ZIP files.
Compare these intervals with the 95% confidence intervals from the chapter. 


Zip, 90% - 5.808
PDF, 90% - 3.496



 7-3  Use the Chap7-MarkovModel.xls MS EXcel workbook to draw graphs
of the probability that exactly P_j (j=0, ..., M) machines are operational
as a function of the ratio lambda/mu. Use M=120 and draw curves for N=2, 5,
and 10.


See:  hw5figs.pdf 


 7-4  Use the Chap7-MarkovModel.xls MS EXcel workbook to generate
a table of MTTR for various values of the ration lambda/mu assuming M=120
and N=5. 


 8-1  Assume that the mix of sessions of type A and B is changed to
f_A = 0.6 and f_B = 0.4.  Use Chap8-CBMG.xls and Chap8-OpenQN.xls to 
recompute the arrival rates for each class and solve the performance
model again for a total arrival rate of sessions, gamma, varying from 8 to
12 session starts/sec. 


Arrival rate	h	s	v	g	c	b

8		0.064	0.263	0.306	0.404	0.571	0.571
9		0.070	0.364	0.373	0.548 	0.794	0.815
10		0.077	0.739	0.584	1.082	1.639	1.757


... and server hits 100% utilization between 10 and 11






 8-8  Given the CBMG of 8.6 and the CPU and disk service demands of
table 8.5, answer the following questions. (Assume 10 new sessions are started
per second.) 
	
	 Find the avg num of visits per session to each of the four e-bus functions.

	
 What is the arrival rate of requests at each of the four e-bus functions?
	
 What is the total utilization of the CPU and of the disk?
	
 What are the residence times at the CPU and disk for each of the four e-bus functions?
	
 What is the response time of each of the four e-bus functions? 
	

See:  hw5figs.pdf 

 10-1  Variation on the random walk through England -- now we have two sons who
behave independently, but are statistically the same 
	
	 What percentage of days is neither son drinking in Leeds?
	
 Once a son finishes a day of kayaking in the Lake District, how long
will it typically be before one of the sons is next there?
	
 How many days each month can the bobbies expect to see one or both 
sons driving to London after drinking in Leeds?
	
 How many days each month does the kayak shop need to have a kayak 
availalbe?
	
See:  hw5figs.pdf 

 10-6  Small store, single worker.  Store can hold at most 2 customers.
Once customer is in store and worker is available, worker spends average of
5 min serving customer.   Worker can serve only 1 customer at a time.
Acceptable service level is that customers spend average of 7.5 minutes in the
store.  What is the maximum allowable rate of customers to the store to 
guarantee this level of service? 

See:  hw5figs.pdf 

 10-11  An ISP has m dial-up ports. Requests arrive at lambda req/sec.
Average session duration is 1/mu (each session completes at rate of mu req/sec. 
	
	 Draw a state transition diagram for the ISP showing the arrival and
departure rates.
	
 Give an expression for the probability, p_k (k=0, ..., m) that
k connections are active as a function of lambda, mu, and m.
	
 Find an expression for the probability p_busy that a customer finds
all dial-up ports busy, as a function of the state probabilities p_k.
	

See:  hw5figs.pdf

Last modified: Wed Dec 3 2008

CSCI 8710 Homework #5

Solution

CSCI 8710
Homework #5