4.6 M/M/s/s Loss Queueing Model

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Leo Anderson
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1 4.6 M/M// Lo Queueig Model Characteritic 1. Iterarrival time i expoetial with rate Arrival proce i Poio Proce with rate 2. Iterarrival time i expoetial with rate µ Number of ervice i Poio Proce with rate µ 3. Multiple Server: Number of erver Idepedet erver Parallel ervice chael Idetical erver Radom choice of idle erver Notatio M / M / / / FCFS 4. Sytem ize i fiite : Total umber i ytem k 5. Queue Diciplie : FCFS OR372-Dr.Khalid Al-Nowibet 1

2 4.6 M/M// Lo Queueig Model Steady-State Ditributio State of the ytem ytem i i tate if there are cutomer i the ytem (waitig or erviced) Let P be probability that there are cutomer i the ytem i the teady-tate., 1, 2, 3,, OR372-Dr.Khalid Al-Nowibet 2

3 4.4 M/M//k Queueig Model Steady-State Ditributio Rate Diagram: Coider a M/M/3/3 ytem e b Queue i cloed ytem ize NO Queue µ µ µ Expoetial Service ytem i tate 1 1 erver buy total ervice peed µ ytem i tate 2 2 erver buy total ervice peed 2µ ytem i tate 3 3 erver buy total ervice peed 3µ OR372-Dr.Khalid Al-Nowibet 3

4 4.6 M/M// Lo Queueig Model Steady-State Ditributio Rate Diagram: 1. Arrival rate for all tate of the ytem 2. Service rate µ if µ 2µ 3µ (-1)µ µ OR372-Dr.Khalid Al-Nowibet 4

5 4.6 M/M// Lo Queueig Model Steady-State Ditributio Balace Equatio: µ 2µ Cut-1 Cut-2 Cut-3 3µ (-1)µ Cut- µ Cut-1 P µp 1 Cut-2 P 1 2µP 2 Cut-3 P 2 3µP Cut- P 1 µp Average Rate out of Cut- Average Rate i to Cut- OR372-Dr.Khalid Al-Nowibet 5

6 4.6 M/M// Lo Queueig Model Steady-State Ditributio Solutio of Balace Equatio: Eq-1 P µp 1 Eq-2 P 1 2µP 2 Eq-3 P 2 3µP Eq- P -1 µp P 1 (/µ) P P 2 (/2µ)P 3 P 3 (/3µ)P P (/µ)p -1 P 1 (/µ) P P 2 ( 2 /2µ 2 )P P 3 ( 3 /6µ 3 )P P ( /!µ )P µ P 1 () P P 2 ( 2 /2)P P 3 ( 3 /6)P P ( /!)P OR372-Dr.Khalid Al-Nowibet 6

7 4.6 M/M// Lo Queueig Model Steady-State Ditributio Solutio of Balace Equatio: Computig P : P + P 1 + P 2 + +P 1 P 1 P + P + ( 2 /2) P + ( 3 /3!)P + + ( /!)P 1 P [ ( 2 /2) + ( 3 /3!) + + ( /!)] 1 P [ (2 /2) + ( 3 /3!) + + ( /!)] 1 OR372-Dr.Khalid Al-Nowibet 7

8 4.6 M/M// Lo Queueig Model Steady-State Ditributio Solutio of Balace Equatio: Computig P : P P fiite um fiite value!!, 1, 2,, Erlag Ditributio! 1 µ NO < 1 P > ad fiite alway for ay, µ ad No Steady-State Coditio o, µ, ad OR372-Dr.Khalid Al-Nowibet 8

9 4.6 M/M// Lo Queueig Model Performace Meaure 1. Blockig Probability BP: BP Pr{ytem i full} P Erlag Lo Formula 2. Effective Arrival Rate e : e arrival rate. Pr{ytem i ot full}. Pr{ < }. (1 BP). (1 P )!! OR372-Dr.Khalid Al-Nowibet 9

10 OR372-Dr.Khalid Al-Nowibet M/M// Lo Queueig Model Performace Meaure 3. Average Cutomer i Sytem L : P L P 1!! 1 1 ( -1)!! ( -1)!! -1 j j 1 j!! ) P (1- Let j 1

11 4.6 M/M// Lo Queueig Model Performace Meaure 4. Average Buy erver L B : L B E[buy erver] E[#Cut. i ervice] L B.P +1.P P P L B L (1 P ) P OR372-Dr.Khalid Al-Nowibet 11

12 4.6 M/M// Lo Queueig Model Performace Meaure 5. Average Cutomer i Queue L q : L q L L B No Queue 6. Utilizatio of the Sytem U: U Pr{ > } P 1 + P 2 + P P 1 P OR372-Dr.Khalid Al-Nowibet 12

13 4.6 M/M// Lo Queueig Model Performace Meaure 7. Utilizatio of the Service SU: SU Pr{ all erver buy } Pr{ } P 8. Average Time Spet i Sytem W : L e.w W L / e 1/µ 9. Average Waitig time i Queue W q : L q e.w q W q L q / e W q / e OR372-Dr.Khalid Al-Nowibet 13

14 4.6 M/M// Lo Queueig Model Example: A cutomer ervice call ceter ha 4 lie to receive cutomer call. Call arrive to the ceter at rate of 4 call per hour. Ay call fid all lie buy will be lot ad the caller may try to call later. The operator ped o average 8 miute to complete the ervice for a caller. It i aumed that the arrival proce i Poio ad the call time i expoetially ditributed. 1. What i the probability that a caller get a buy igal? 2. O average how may lie are buy? 3. What i the probability that lie 2 i buy? 4. What i the probability that 2 lie are buy? 5. What i the probability that a caller wait? 6. What i the average umber lie that are ot buy? 7. What i the average umber of lot call durig the day (work hor 7: am 5: pm)? OR372-Dr.Khalid Al-Nowibet 14

15 4.6 M/M// Lo Queueig Model Example: Arrival: 2 cut/hr Poio Service: E[S] 12 mi. Expoetial µ 1/E[S] 1/12 cut/mi 6/12 5 cut/hr Number of Server: 3 Sytem Size waitig eat + ervice eat M/M/3/1 / 2/(3(5)) 1.33 OR372-Dr.Khalid Al-Nowibet 15

16 OR372-Dr.Khalid Al-Nowibet M/M// Lo Queueig Model Example: 1. If you arrive to the hop, what i the probability that you get erved? Pr{get erved} Pr{eter} Pr{ < 1} 1 P 1 Pr{get erved} !! P + k P! P P! P < k P

17 4.4 M/M//k Queueig Model Example: 2. O average how may haircut i performed per hour? Average umber of hair cut i performed per hour e (1 P 1 ) 2 (.7315) haircut/hr. 3. If you eter the hop at 9: am whe do you expect to leave? E[Departure time] 9: + W W L / e (P )/ e 7.62/ hr :32 E[Departure time] 9: + W 9: + :32 9:32 OR372-Dr.Khalid Al-Nowibet 17

18 4.4 M/M//k Queueig Model Example: 4. What i the probability that a cutomer wait if he eter? Pr{waitig} Pr{ 3 < 1} P 3 + P P What i the average empty eat i the hop? E[# empty eat] waitig eat E[# waitig cutomer] L q L e /µ (14.684)/ E[# empty eat] eat OR372-Dr.Khalid Al-Nowibet 18

19 4.4 M/M//k Queueig Model Example: 6. The ower of the hop decided to accept ay arrival to eter the hop, what i the miimum barber he mut hire? Accept ay arrival ytem ize k The ew ytem i M / M / / mut be le tha 1 4 4/ < 1 > 4 5, 6, 7,... miimum umber 5 baraber OR372-Dr.Khalid Al-Nowibet 19

Queueing Theory (Part 3)

Queueing Theory (Part 3) Queueig Theory art 3 M/M/ Queueig Sytem with Variatio M/M/, M/M///K, M/M//// Queueig Theory- M/M/ Queueig Sytem We defie λ mea arrival rate mea ervice rate umber of erver > ρ λ / utilizatio ratio We require