MSG 253 Queueing Systems and Simulation [Sistem Giliran dan Simulasi]

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1 UNIVERSITI SAINS MAAYSIA Secod Semester Examiatio 23/24 Academic Sessio Jue 24 MSG 253 Queueig Systems ad Simulatio [Sistem Gilira da Simulasi] Duratio : 3 hours [Masa : 3 jam] Please check that this examiatio paper cosists of THIRTEEN pages of prited material before you begi the examiatio. [Sila pastika bahawa kertas peperiksaa ii megadugi TIGA BEAS muka surat yag bercetak sebelum ada memulaka peperiksaa ii.] Istructios : [Araha : Aswer all three [3] uestios. Jawab semua tiga [3] soala.] I the evet of ay discrepacies, the Eglish versio shall be used. [Sekiraya terdapat sebarag percaggaha pada soala peperiksaa, versi Bahasa Iggeris hedaklah digua pakai]. 2/-

2 -2-. (a) A ueueig process that is based o the birth ad death process has the followig birth ad death rates: ( +), for,, 2, 3,., for, 2, 3,. Draw a rate-diagram to represet the ueueig system. (ii) Show that P! e, where. (iii) Next, show that - + e. [5 marks] (b) Cosider a sigle server ueueig system where some customers balk (refuse to eter the system) ad some customers who eter the system later get impatiet ad reege (leave without beig served). Customers arrive accordig to a Poisso process with a mea rate of 4 per hour. A arrivig customer who fids customers already there will balk with the followig probabilities: { } P balk already there if,.5 if,.75 if 2, if 3. Service times have a expoetial distributio with a mea of hour. A customer already i service ever reeges, but the customers i the ueue may reege. I particular, the remaiig time that the customer at the frot of the ueue is willig to wait i the ueue before reegig has a expoetial distributio with a mea of hour. For a customer i the secod positio i the ueue, the time that she or he is willig to wait i this positio before reegig has a expoetial distributio with a mea of 2 hour. (ii) (iii) (iv) Costruct the rate diagram for this ueueig system. Obtai the steady state distributio of the umber of customers i the systems. Fid the expected fractio of arrivig customers who are lost due to balkig. Fid the expected umber of customers i the system. [5 marks] 3/-

3 -3-. (a) Suatu proses gilira yag beasaska kepada proses lahir da mati mempuyai kadar kelahira da kematia seperti berikut: ( +), bagi,, 2, 3,., bagi, 2, 3,. ukiska gambar rajah kadar yag mewakili sistem gilira itu. (ii) Tujukka bahawa P (iii)! e, di maa. Kemudia, tujukka bahawa - + e. [5 markah] (b) Pertimbagka suatu sistem gilira satu pelaya dega sebahagia daripada pelaggaya egga (tidak memasuki sistem) da sebahagia daripada pelagga yag telah memasuki sistem pula mejadi tidak sabar da belah (meiggalka sistem sebelum dilaya). Ketibaa pelagga berlaku megikut proses Poisso dega kadar mi 4 sejam. Seorag pelagga yag tiba da medapati seramai pelagga sudah berada di saa aka egga dega kebaragkalia berikut: { } P egga sudah berada di saa jika,.5 jika,.75 jika 2, jika 3. Masa laya adalah megikut agiha ekspoe dega mi jam. Seorag pelagga yag sedag dilaya tidak aka belah, tetapi pelagga yag sedag meuggu mugki aka belah. Secara khususya, masa yag tiggal utuk terus meuggu sebelum belah bagi pelagga yag berada palig hadapa barisa meuggu adalah megikut agiha ekspoe dega mi jam. Bagi pelagga yag berada di tempat kedua barisa meuggu pula, masa yag tiggal utuk terus meuggu sebelum belah adalah megikut agiha ekspoe dega mi 2 jam. (ii) (iii) (iv) Betukka gambar rajah kadar bagi sistem gilira ii. Dapatka agiha bilaga pelagga di dalam sistem dalam keadaa matap. Tetuka isbah jagkaa pelagga yag tiba tetapi hilag disebabka oleh keeggaa. Tetuka bilaga jagkaa pelagga di dalam sistem. [5 markah] 4/-

4 -4-2. (a) Cars arrive at a small petrol statio to refuel accordig to a Poisso process with rate 3 per hour, ad have a expoetial service time distributio with mea 4 miutes. Sice there are four petrol pumps available, four cars ca refuel simultaeously, but ufortuately there is o room for cars to wait. Hece, if a car arrives whe all pumps are busy, the driver leaves immediately. For each customer that is served, a average profit is made of RM7. (ii) (iii) Determie the probability that a arrivig car is ot refueled. What is the expected umber of pump that is beig used? What is the expected profit per day (cosistig of eight hours)? Suppose that the statio maager has the opportuity to buy a adjacet parkig lot. So there is room to wait for all cars that arrive whe the four pumps are busy. Suppose all drivers decide to wait istead of leavig whe this happes. (iv) (v) (vi) (vii) Determie the probability that a arrivig car has to wait. What is the expected umber of cars that will occupy the adjacet parkig lot? How log o average will a car occupy the adjacet parkig lot? What is the expected profit per day i this case? [4 marks] (b) Past records idicate that each of the 5 laser computer priters at the School of Mathematical Scieces, USM, eeds repair after about 2 hours of use. Breakdows have bee determied to be Poisso distributed. The oe techicia o duty ca service a priter i a average of 2 hours, followig a expoetial distributio. Priter dowtime costs RM2 per hour. Techicias are paid RM25 per hour. Should the School hire a secod techicia? [3 marks] (c) A state agecy that hadles compesatio for those who are uemployed is cosiderig two optios for processig applicatios. Optio : Four clerks process applicatios i parallel from a sigle ueue. Each clerk fills out the reuired form i the presece of the applicat based o iformatio that is verbally related to the clerk. Processig time is expoetially distributed with a mea of 45 miutes. Optio 2: Each applicat first fills out the form without the help of the clerk. The time reuired to accomplish this is expoetially distributed, with a mea of 65 miutes. Whe the applicat fiishes the form, he or she jois a sigle ueue to await a review by oe of the four clerks. The time reuired to review a form is expoetially distributed with a mea of 5 miutes. Give that the arrival of applicats is a Poisso process with a mea rate of 4.8 per hour, compare the two optios with respect to the expected umber of applicats i the system ad the expected time i the system. [3 marks] 5/-

5 -5-2. (a) Kereta tiba di sebuah stesye petrol kecil utuk megisi petrol megikut proses Poisso dega kadar 3 sejam, da masa laya adalah megikut agiha ekspoe dega mi 4 miit. Oleh keraa terdapat empat pam petrol sahaja, empat kereta dapat megisi petrol dega seretak. Malagya, tidak ada ruag bagi kereta lai utuk meuggu. Dega itu, jika sesebuah kereta tiba semasa kesemua pam sibuk diguaka, pemadu kereta berkeaa terpaksa meiggalka stesye itu dega segera. Bagi setiap pelagga yag dilaya, purata keutuga yag diperolehi adalah RM7. (ii) (iii) Tetuka kebaragkalia bahawa sesebuah kereta yag tiba tidak dapat diisi petrol. Berapakah bilaga purata pam yag diguaka? Berapah keutuga jagkaa haria (melibatka lapa jam)? Kataka pegurus stesye berpeluag membeli lot parkir bersebelaha. Dega itu, ada ruag meuggu bagi kesemua kereta yag tiba semasa keempat-empat pam sedag diguaka. Kataka kesemua pemadu ada meuggu apabila keadaa ii berlaku. (iv) (v) (vi) (vii) Tetuka kebaragkalia bahawa sesebuah kereta yag tiba terpaksa meuggu. Berapakah bilaga jagkaa kereta yag ada berada di lot parkir bersebelaha? Berapa lamakah pada purataya sesebuah kereta aka berada di lot parkir bersebelaha? Berapakah jagkaa keutugga haria bagi kes ii? [4 markah] (b) Rekod yag lepas meujukka bahawa setiap satu daripada 5 pecetak computer laser di Pusat Pegajia Sais Matematik, USM, perlu dibaiki selepas 2 jam pegguaa. Kerosaka didapati berlaku megikut agiha Poisso. Seorag jurutekik yag bertugas boleh membaiki pecetak dalam masa purata 2 jam, megikut agiha ekspoe. Kos ketidak-operasia pecetak adalah RM2 sejam. Jurutekik dibayar RM25 sejam. Patutkah Pusat Pegajia megupah seorag lagi jurutekik? [3 markah] (c) Sebuah perbadaa kerajaa egeri yag megedalika urusa subsidi kepada mereka yag megaggur sedag meimbagka dua opsye utuk memproses permohoa. Opsye : Empat kerai aka memproses permohoa secara selari daripada satu barisa meuggu. Setiap kerai aka megisi borag utuk pemoho berasaska kepada maklumat yag diberika secara lisa kepadaya. Masa pemprosesa adalah megikut agiha ekspoe dega mi 45 miit. Opsye 2: Setiap pemoho megisi sediri borag yag disediaka. Masa yag diperluka ialah megikut agiha ekspoe dega mi 65 miit. Apabila borag siap diisi, pemoho aka memasuki satu barisa meuggu utuk meuggu gilira peyemaka borag oleh salah seorag daripada empat kerai yag ada. Masa peyemaka borig ialah megikut agiha ekspoe dega mi 5 miit. Jika ketibaa pemoho adalah megikut proses Poisso dega kadar mi 4.8 sejam, badigka dua opsye itu berasaska kepada bilaga jagkaa pemoho di dala sistem da jagkaa masa di dalam sistem. [3 markah] 6/-

6 -6-3. (a) To facilitate customers with oly check cashig eeds, a certai bak has assiged oe special teller to perform oly this service. About 4% of arrivig customers are of this ature. The remaiig 6% have additioal trasactios to process ad must use oe of the other four tellers i the bak. Customers arrive at itervals described by a uiform distributio with a mea of 66 secods ad a rage of ± secods, ad either joi the check-cashig teller lie or the sigle lie for the other four tellers. Check cashig reuires 2 ± 3 secods, ad all other back busiess takes 36 ± 2 secods per customer. (ii) (iii) Write a GPSS World program to simulate a 7-hour bakig day from 9:am to 4:pm. Measure the statistics for the ueues. For the arrival of those customers who have oly check cashig eeds, if the special teller is busy ad oe of the four regular tellers is free, these customers are allowed to go to the regular tellers. How do you modify the GPSS program i for this? Suppose that upo the arrival of those customers who eed the regular service, if the four regular tellers are busy ad the special teller is free, these customers are allowed to go to the special teller, how do you modify your GPSS program i (ii)? [5 marks] (b) Ships arrive at a harbor at a iter-arrival time of either, 2 or 3 hours, each with eual chaces of happeig. There are three berths to accommodate them. They also eed the service of a crae for uloadig ad there are two craes. After uloadig, % of the ships stay to refuel before leavig; the others leave immediately. Ships do ot eed the craes for refuelig. The uloadig time is either 2, 3, 4 or 5 hours with probabilities of.,.3,.4 ad.2, respectively. The refuelig time is either, ½, or 2½ hours, each with eual chaces of happeig. The aswers to the followig uestios are to be provided: i) What is the average time that a ship spet at the harbor? ii) What proportio of time are the craes idle? iii) What is the average time that a ship spet waitig for a berth to be available? (Use the eclosed 2-digit radom umber table with the first colum for iter-arrival time, secod colum for uloadig time, the third colum for determiig whether to refuel or ot ad the fourth colum for refuelig time.) [5 marks] 7/-

7 -7-3. (a) Bagi memudahka pelagga yag haya berurusa meuaika cek, sebuah bak telah megumpukka satu teller khas haya utuk urusa berkeaa. ebih kurag 4% daripada pelagga adalah di dalam kategori ii. ebiha 6% pelagga memerluka urusa lai utuk diproses da mereka mesti megguaka salah satu daripada empat teller lai yag disediaka. Pelagga tiba megikut selag yag bersifat agiha seragam dega mi 66 saat da berjulat ± saat, da mereka memasuki sama ada barisa meuggu utuk teller meuaika cek ataupu satu barisa meuggu utuk empat teller yag lai. Meuaika cek memerluka 2 ± 3 saat, da semua urusa lai memerluka 36 ± 29 saat bagi setiap pelagga. Tuliska satu aturcara GPSS World utuk mesimulasi satu hari bekerja bak selama 7 jam bermula 9: pagi sehigga 4: petag. Tetuka statistik bagi semua barisa meuggu. (ii) Bagi pelagga yag tiba utuk urusa meuaika cek sahaja, jika teller khas sedag sibuk da salah satu daripada empat teller biasa adalah berseag, mereka dibearka berurusa dega teller biasa. Bagaimaakah aturcara GPSS di dalam perlu diubahsuai utuk ii? (iii) Kataka bahawa semasa ketibaa pelagga yag memerluka urusa biasa, jika keempat-empat teller biasa adalah sibuk da teller khas sedag berseag, mereka dibearka berurusa dega teller khas itu. Bagaimaakah aturcara GPSS di dalam (ii) perlu diubahsuai utuk ii? [5 markah] (b) Kapal tiba di pelabuha dega masa atara ketibaa ialah sama ada, 2, atau 3 jam, dega setiap satuya sama mugki aka berlaku. Tiga pelatar disediaka utuk keguaa kapal. Kapal juga memerluka kre utuk pemuggaha da terdapat dua kre di perlabuha itu. Selepas siap pemuggaha, % daripada kapal aka terus berlabuh utuk megisi miyak; yag laiya aka terus beredar. Semasa pegisia miyak, kre tidak diperluka. Masa memuggah ialah sama ada 2, 3, 4 atau 5 jam dega kebaragkalia masig-masigya ialah.,.3,.4 da.2. Masa utuk megisi miyak pula ialah, ½ atau 2½ jam, dega setiap satuya sama mugki aka berlaku. Jawapa kepada soala-soala berikut mestilah diberika: i) Berapakah masa purata sesebuah kapal berada di pelabuha? ii) Berapakah peratusa masa berseag kre? iii) Berapakah masa purata sesebuah kapal meuggu utuk megguaka pelatar? (Guaka sifir ombor rawak 2-digit yag disertaka dega lajur pertama utuk masa atara ketibaa, lajur kedua utuk masa memuggah, lajur ketiga utuk meetuka sama ada perlu megisi miyak ataupu tidak da lajur keempat utuk masa megisi miyak) [5 markah] 8/-

8 -8- APPENDIX / AMPIRAN Formulas for Queueig Theory:. M/M/: r / ( ) P r r for,, 2,... 2 [ ] ( ) W, W Pw> t e t/ w P w > t re t/ w ( ) 2. M/M/s: s P ( ) ( / )! ( / ) s s ( / ) s! ( ) P, if s 2 P, if s W, W W + / + / P ( ) s / s / P + s! ( )! ss! s P t( s / ) t e P w t e > + s! ( ) s / s ( ) { } P w t > P w e { } s whee P w P t 9/-

9 -9- APPENDIX 2 / AMPIRAN 2 3. M/M/s: fiite populatio of size M. s M M M! P s + s s s! M P, if s M! P P, if s M s < s s!, if > M s M M M! P + s s s s! s M s+ P ( s ) W, W ( M ) ( M ) 4. M/G/: 5. M / E k /: P σ ( ) W, W w + W 2 + k 2k ( ) + k 2k W W + / W ( ) /-

10 -- APPENDIX 3 / AMPIRAN 3 6. M/M//k: P ( ) k + k + ( ) ( ) For k ( k+ ) + k k + ( )( ) k+ k ( ) ( P ) k + ( ) W /, W W / / For k 2 7. M/M/s/k: P s! P P s s k s s! ( < ) ( ) k s + s s ( / ) s + for! s! s P s s s ( / ) + ( k s+ ) for! s! s S ( sr) ( r ) P r 2 r ( r)( k s+ ) r s! k s+ k s /-

11 -- APPENDIX 4 / AMPIRAN 4 8. M/M/s/s: + s P s ( s )( s)! W, W W W ( ) ( sr ) i ( sr ) ( P ) / /! P for s s i /! i i s k ( ) / s! Ps where. s r i s /! i Ps W Ps ( ), where ( ) 9. M / M / : / ( / ) e P! for,,2,... / W 2/-

12 -2- APPENDIX 5 / AMPIRAN 5. M/M/: state-depedet service ( k) ( k) k k P ( ) ( ) ( ) ( ) k k+ ( /, / ) ( ) ( ) ( ) k k k + k k k k P P + < W W P W W + P < k P P k. M/M/: fiite populatio of size M. M M! P ( M )! M! P P for,2,..., M ( M )! M [ P ] + M ( P ) W, W where M ( ) 3/-

13 -3- APPENDIX 6 / AMPIRAN 6 TWO-DIGIT RANDOM NUMBER TABE ooo O ooo -

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