page 1 This question-paper you hand in, together with your solutions. ================================================================

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1 age EXAMINATION Jan 9 Time :5-7:5 QUEUEING THEORY EP9, HF ( 6H78), Lecturer: Armin Halilovic Instructions:. You ARE allowed to use a calculator.. You are NOT allowed to use your own tables of mathematical formulas on the exam.. Use of any communication device when taing this examination is strictly rohibited. 4. Write your name and ID-number on each age. 5: Write the question number on each age. 6: Start a new question on a new age. Grading: For each correct solution, four oints will be awarded. Credit will be given or resentation and methods of solutions. A maximum of oints (8+ bonus oints) can be earned. Points can also be deducted for unsubstantiated answers. At least 6 oints are required to ass the exam. Grading scale: The total number of ossible oints is. 8 - oints are required for an A grade; 5-7 oints are required for a B grade; -4 oints are required for a C grade; 9 - oints are required for a D grade; 6-8 oints are required for an E grade; 4-5 oints result in a Fx ; - oints result in a F grade. This question-aer you hand in, together with your solutions. Question ) (4 oints) We model a queueing system as a birth-and-death rocess with two states E and E with arrival rate λ and service rate μ 7 (see Fig ). Let ( t ) be the robability that the system is at state Fig E and let ( t ) be the robability that the system is λ at state E. We assume that (). 4 Determine the differential equation for ( t ). E E Use the Lalace transform to solve the equation and find ( t ). ( If you solve the equation using some other method instead of Lalace then you get max μ oints for this question) age

2 age Question ) (4 oints) A system can be modeled as a three state Marov chain. The system has states E, E and E with transition rates defined in the rate transition diagram (Fig a) i) ( oin Find the rate transition Q- matrix. Fig a E E 4 E ii) ( oints) Find the stationary robability vector. iii) ( oin Determine the system, with differential equation and linear equation, for the time-deendent (transien robabilities, and ( t ). (You do not need to solve the diff eq. system iii.) Question. a) ( oints) Derive the following formulas for an M/M/queueing system: a), a) N, b) ( oints). A communication channel has the caacity of K bits/second. Data acets arrive at the channel according to a Poisson rocess with rate λ6 acets er minute. The acets have an exonentially distributed length with a mean of v 4 bits. We assume that the channel can be modeled as an M/M/ system with queueing disciline FCFS (First- Come- First- Served). b. Determine the minimum value of K which ensure that the average total time T, sent in the system by a customer, is less than or equal to T.5 seconds. b. For that K find the robability that the time sent in the system by a customer is less than or equal to.6 seconds and greater than. seconds. Question 4. A telehone system can be modeled as an M/M/4/ wait system with the arrival rate of 5 calls er minute and the service time of x minute. a) Draw the rate transition diagram b) Find the offered, effective and bloced traffic. c) find the blocing robability, that is the robability for an arriving call to be bloced. Question 5. We consider an M/M/// system ( servers waiting laces, customers) with arrival rate from one customer β arrivals er minute and service time x 6 seconds. a) ( oints) Find,,..., 5 b) ( oin Find the robability that an arriving customer find 5 customers in the system. age

3 age Question 6. Find x, E ( ~ x ), μ, and then use the Pollacze - Khinchin formula, to find W for an M/G/ system where λ /5 acets er ms, and π cost t f othervise is the density function of the random variable ~ x (time unit ms). Hel for question 7: π / ( ~ x E x) tf dt, E x ) ( ~ π / t f dt Pollacze - Khinchin formula λ E( ~ x ) W. ( ) Question 7. We consider a queueing net with three M/M/ queues as shown in Fig. 8. Three servers S, S and S wor with service rates μ 4 rograms /min, μ 5 rograms/min, resective μ 5 rograms /min. New rograms (customers) arrive at a servers S and S according to Poisson rocesses of rate λ Α, and λ Β rograms er minute. After getting service the rograms continue according to Fig 8. Find the average number of the rograms in the queueing net (NN+N+N). Fig 8. λ Β λ Α Q μ 75% Q S μ 5% 5% Q S % 5% out μ 8% out Good luc! age

4 age 4 Solutions: Question ) (4 oints) Question ) (4 oints) We model a queueing system as a birth-and-death rocess with two states E and E with arrival rate λ and service rate μ 7 (see Fig ). Let be the robability that the system is at state Fig E and let ( t ) be the robability that the system is λ at state E. We assume that (). 4 Determine the differential equation for. E E Use the Lalace transform to solve the equation and find. ( If you solve the equation using some other method instead of Lalace then you get max μ oints for this question) Solution: a) Q 7 7 System: + 7 ( + ( Substitution + ( gives the following differential equation + 7 ( eq) with the initial value: (). 4. Using the Lalace transform we get: 7 sp () P + s 7 sp.4 P + s 7 sp + P.4 + s.4 7 P + s + s( s + ) From the formula table we get t t t e + e.4e + 7 t 4 t e + e t e 7 t Answer: ( e age 4

5 age 5 Question ) (4 oints) A system can be modeled as a three state Marov chain. The system has states E, E and E with transition rates defined in the rate transition diagram (Fig a) i) ( oin Find the rate transition Q- matrix. Fig a E E 4 E ii) ( oints) Find the stationary robability vector. iii) ( oin Determine the system, with differential equation and linear equation, for the time-deendent (transien robabilities, and (. (You do not need to solve the diff eq. system iii.) Solution: i) Q r Let (,, ) denote the stationary robability vector. r From Q r we get the following system Solving this system we get,, Answer ii) The stationary robability vector is (,, ) Answer iii) r r From Q we get the following system ( + ( ( ( + + age 5

6 age 6 Question. a) ( oints) Derive the following formulas for an M/M/queueing system: a), a) N, b) ( oints). A communication channel has the caacity of K bits/second. Data acets arrive at the channel according to a Poisson rocess with rate λ6 acets er minute. The acets have an exonentially distributed length with a mean of v 4 bits. We assume that the channel can be modeled as an M/M/ system with queueing disciline FCFS (First- Come- First- Served). b. Determine the minimum value of K which ensure that the average total time T, sent in the system by a customer, is less than or equal to T.5 seconds. b. For that K find the robability that the time sent in the system by a customer is less than or equal to.6 seconds and greater than. seconds. Solution: a) From the rate transition diagram we have. a) The average number of the customers in the system can be calculated using the formula for the exectation of a discrete random variable: N (*) If we differentiate with resect to the formula for the sum of a geometric series we obtain (**) ( ) We substitute (**) in (*) and get N ( ) which roves the formula. ( ) age 6

7 age 7 Solution b) First λ6 acets er minute acets er second. Using T we get the following equation μ λ μ λ μ μ Now, K v μ Answer b) K 48 bits er sec. b)since, for M/M/, P( a < ~ s b) e ( μ λ ) a F F ( μ λ ) t ~ s P s e, we have ( ~ ( μ λ ) b ( μ λ ) a ~ s ( b) F~ s ( a) ( e ) ( e ) ( μ λ ) b e ~.6 P (. < s.6) e e. Thus: Answer b) e.6 e..476 Question 4. A telehone system can be modeled as an M/M/4/ wait system with the arrival rate of 5 calls er minute and the service time of x minute. a) Draw the rate transition diagram b) Find the offered, effective and bloced traffic. c) find the blocing robability, that is the robability for an arriving call to be bloced. Answer 4. a) age 7

8 age 8 b) The offered traffic 5. The effective traffic.59 The bloced traffic.4677 P blocing robability.954 b Question 5. We consider an M/M/// system ( servers waiting laces, customers) with arrival rate from one customer β arrivals er minute and service time x 6 seconds. a) ( oints) Find,,..., 5 b) ( oin Find the robability that an arriving customer find 5 customers in the system. a) b) r λ λ.7 mean Question 6. Find x, E ( ~ x ), μ, and then use the Pollacze - Khinchin formula, to find W for an M/G/ system where λ /5 acets er ms, and age 8

9 age 9 π cost t f othervise is the density function of the random variable ~ x (time unit ms). Hel for question 7: π / ( ~ x E x) tf dt, E x ) ( ~ Pollacze - Khinchin formula Solution: λ, 5 π / π / x E( ~ x) tf dt t costdt E ( ~ x π / ) t f dt t π / μ x π π λx Hence λ E( ~ x ) W. 576 ( ) π / t f dt λ E( ~ x ) W. ( ) costdt π / π π [ t sin t + cost] π / π [ t sin t + t cost sin t].4674 Question 7. We consider a queueing net with three M/M/ queues as shown in Fig. 8. Three servers S, S and S wor with service rates μ 4 rograms /min, μ 5 rograms/min, resective μ 5 rograms /min. New rograms (customers) arrive at a servers S and S according to Poisson rocesses of rate λ Α, and λ Β rograms er minute. After getting service the rograms continue according to Fig 8. Find the average number of the rograms in the queueing net (NN+N+N). Fig 8. 4 age 9

10 age λ Β λ Α Q μ 75% Q S μ 5% 5% Q S % 5% out μ 8% out Lyca till! Solution: The effective (ne arrival rates into these two queues are λ +.5λ +.λ λ +.75λ λ.5λ From the system we get: λ 5 λ 5, λ 5 Since μ 4, μ 5 and μ 5 we have λ λ, μ μ λ and μ N, N, N 5 N N + N + N.5 age