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COM6515 NETWORK PERFORMANCE ANALYSIS
创建日期：2024-04-12 16:10:46
标签： COM6515
NETWORK PERFORMANCE ANALYSIS

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COM6515 Data Provided: 

NONE DEPARTMENT OF COMPUTER SCIENCE 

NETWORK PERFORMANCE ANALYSIS 

2 hours ANSWER ALL QUESTIONS. All questions carry equal weight. Figures in square brackets indicate the percentage of avail- able marks allocated to each part of a question. Registration number from U-Card (9 digits) — to be completed by student COM6515 1 TURN OVER COM6515 THIS PAGE IS BLANK COM6515 2 CONTINUED COM6515 1. a) Consider a steady state queue in which the number of servers is unbounded. (i) Write down the equations that define the arrival rate and service rate in terms of the state k. Explain the differences between them. [15%] (ii) Use the steady state balance equations to derive the steady state probability that the system is in state k. [30%] (iii) What is the average number of people in the system and what is the average delay? [10%] b) The system now changes to a finite number of servers m, and a customer is not allowed to wait. Thus if a customer arrives and finds all the service tills busy, the customer leaves the system. (i) What are the service and arrival rates for this system? [20%] (ii) Calculate the blocking probability. [25%] COM6515 3 TURN OVER COM6515 2. a) Consider an M/M/m queue in which there is one queue and m servers. The steady state probabilities for this system are Pk =   P0 ( (mρ)k k! ) k < m P0 ( mmρk m! ) k ≥ m , ρ = λ mµ . (i) What do λ and µ represent? Also, state the restriction on the value of ρ. [15%] (ii) Derive an expression for P0 in terms of m and ρ. Simplify the expression as much as possible. [20%] b) Consider a switch in a computer network that has one input port and three output ports, and is modelled as an M/M/3 queue. (i) Write down the expressions for the probabilities Pk for k < 3 and k ≥ 3, and the restriction on the value of ρ. [10%] (ii) Show that [15%] P0 = [ 1+3ρ+ 9 2 ρ2 + 9ρ 3 2(1−ρ) ]−1 . (iii) Show that the average number of packets in the system is P0 2 ∑ k=0 k(3ρ)k k! + 9P0 2 ∞ ∑ k=3 kρk and that this expression simplifies to P0 [ 9ρ 2(1−ρ)2 − 3ρ 2 ] . [40%]

 COM6515 4 CONTINUED COM6515 3. a)

 The Poisson process is the arrival process that is most frequently used to

 model the behaviour of queues. (i) Derive expressions for the mean and variance of the Poisson distribution 

at a specific time in terms of the rate λ. [25%] (ii) What is the probability that there are no arrivals in the time interval T ? [5%] 

(iii) What is the probability that there is at least one arrival in the time interval T ? [5%] 

b) Consider an M/M/1 queue for which the arrival and service rates at state k are λk = λαk, k ≥ 0, 0 ≤ α < 1 µk = µ, k ≥ 1 (i) 

Calculate the probability Pk that there are k customers in the system. Express your answer in terms of P0. [30%] 

(ii) Deduce an expression for P0 and calculate the probability that there are two or more people in the system. [15%]

 (iii) Show that if λµ < 1, then P0 > 1− λ µ [10%] (iv) Is the condition λµ < 1 necessary for a steady state solution to exist?

 Can this solution exist for λµ ≥ 1? Explain your answer. [10%] END OF QUESTION PAPER COM6515 5

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MATH20602 COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 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MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A 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