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Unformatted text preview: SG/u‘i/osﬁ Name: ISEN316 TEST 1 2262009 WHITE Instructions: Use your own paper, put your problems in their order, staple this Test sheet to the front. Put your name and UN number of this sheet. Use only one side of the solutions papers (the backs will not be graded). Show your work for credit. Compute
your answers to at least 3 decimals. 30 points . I. Consider a workstation model with an Erlang2 service process. To model this
situation using the statediagram approach, this service process is broken into two
exponential phases (denoted below as p1 and p2). So,‘ assume that for an M / E2 / 1/ 3 model, we have solved for the probabilities:
Po: Pinup Pun: Pap” F2422: pap” p3,p2 where pm stands for 1' customers in the system and the server process in phase It
denoted as pic (I: = 1, 2) . Answer the following questions using the above probabilities: (a) write the equation for the server utilization;
(b) write the equation for the MR, (note in the queue);
(6) write the equation for the proportion of the time that theserver is in phase 2; (d) write the equation for the system throughput ths; 20 points 
2. Estimate the cycle time in the queue, CI}, for the following model (E2 / G I 2/ co ). The data for the interarrival and service processes are: '
mean arrival rate is 10 j obs/hr. 
mean job processes time is 10 min.
variance of the job processing time (service) is 25 minz. 30 points
3. Give a short concise deﬁnition for each of the following:
(a) mean
(1)) squared coefﬁcient of variation
(0) symbol M
(of) symbol E4
(2) how are (a) and (1)) related?
0‘) in the shorthand notation (by Kendall): A l B l C I D
deﬁne ﬁeld “A”
(g) in the shorthand notation (by Kendall): A l B l C l D
deﬁne ﬁeld “C”
(h) deﬁne cycle time
(1') deﬁne the relationship between CTS and CI},
(J) Little‘s Law applies to (pick one)
(I) factory as a whole, (2) workstations, (3) both 20 points
4. Consider an Erlang~2 arrival process (with mean interarrival rate 2.) and a two phase
service process with the ﬁrst phase’s mean rate being y and the second phase’s mean . rate being y. There is only one server (with two phases) and a maximum of 3 customers are allowed into the systemat the same time. This is a speciﬁc model of the general
notational structure: E2 1' G / 1/ 3 . (0) Deﬁne the appropriate states; (b) Develop a state diagram with nodes representing states and arcs representing
ﬂows between states, and label the nodes and arcs with the appropriate
information. ' JP! + ﬁ3ﬂ+ P2350! ﬂaFLWLng +1214”;
. : If??? 2/0/45: 4.1: 5095» kg was;
: chjvgsz)(l {Au 5533 ( 5 .» .. FA»;
‘%’¥")(%Gﬂ ——— 0 W cf) : cgrE’ZEJ
3) beﬂ Se/u‘bow Name: UlN: ISEN316 TEST 1 2262009 YELLOW Instructions: Use your own paper, put your problems in their order, staple this Test sheet
to the front. Put your name and UIN number of this sheet. Use only one side of the solutiOns papers (the backs will not be graded). Show your work for credit. Compute
your answers to at least 3 decimals. 20 points
1. Consider a two phase (general) arrival process (with mean phase rates it and y) and
an Erlang2 service process with the mean rate it .. There is only one server and a maximum of 4 customers are allowed into the system at the same time. This is a speciﬁc
model of the general notational structure: G / E2 11! 4. (a) Deﬁne the appropriate states;
(b) Develop a state diagram with nodes representing states and arcs representing ﬂows
between states, and label the nodes and arcs with the appropriate information. 30 points
2. Give a short concise deﬁnition for each of the following:
(a) variance
(5) squared coefficient of variation
(c) symbol G
(d) symbol E3
(3) how are (a) and (b) related? '
(f) in the shorthand notation (by Kendall): A/B / C l D
deﬁne ﬁeld “B” (g) in the shorthand notation (by Kendall): A l B l C l D
I define ﬁeld “D”
(It) deﬁne cycle time in the queue ‘
(1') deﬁne the relationship between CT 5 and CTq '
(j) Little’s Law applies to»(pick one)
(i) factory as a whole, (2) workstations, (3) both 30 points
3. Consider a workstation model with a twophase arrival process. To model this
situation using the statediagram approach, this arrival process is broken into two exponential phases (denoted below asp] and 192). So, assume that for a E2 / M l 2/ 4
model, we have solved for the probabilities: Pepi, Palm Plum: P'va Pam: P2,,o2: P3431! P3.p29p4,p1’ P4.p2 where pm stands for 1' customers in the system and the arrival process in phase It denoted as pk (k = l, 2) . Answer the following questions using the above probabilities: (a) write the equation for the server utilization;
(3;) write the equation for the WIPq (note in the queue); (.3) write the equation for the proportion of the time that the arrival process is in
phase 2; (d) write the equation for the system throughput tbs; 20 points
4. Estimate the cycle time in the queue, CE}, for the following model (E3 / GI 2/ oc ). The data for the interarrival and service processes are:
mean arrival rate is 12 jobs/hr.
mean job processes time is 8 min.
variance of the job processing time (service) is 36 minz. 5) CT; == cg+5€7§7 (33W ,4
8 a) : Egan?! +P’JP'2)+ (?§£PLJPJ+ riff—2.)
.7 ‘5) WIP? 2 Mg?“ +gin3 +9Cﬁbﬁ+ 43%;)
4
80¢) 30%;
7 4) 17% = A " “RX 3395392 ...
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 Spring '08
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