Basic definitions and relations

Transcription

1 Basic definiions and relaions Lecurer: Dmiri A. Molchanov hp://

2 Kendall s noaion for queuing sysems: Arrival processes; Service ime disribuions; Examples. Lile s resul: Basic resul; Proof; Example. Lecure: Basic definiions and relaions 2

3 1. Kendall s noaion for queuing sysems exponenial arrivals exponenial service Erlang arrivals hyperexponenial service infinie # of waiing posiions limied # of waiing posiions exponenial service general arrivals general service general arrivals K servers infinie # of waiing posiions infinie # of waiing posiions Figure 1: Differen ypes of queuing sysems. Quesion: how o represen i in a compac form? Lecure: Basic definiions and relaions 3

4 Which elemens we have o describe: arrival process; service ime disribuion; how many servers does he sysem have; are here waiing posiions in he sysem; if yes, how many waiing posiion does he sysem have; wha is he service discipline; are here special rules applicable o sysem: are here vacaions of he server; do he arrivals occur in baches or no; are here prioriies in service for cerain cusomers. Noe: we should say a leas he class of arrivals, deparures, # of servers... Lecure: Basic definiions and relaions 4

5 A simple noaion of queuing sysem: Kendall in 60 h ; proposed o use a leers o describe characerisics of queuing sysem; leers are separaed by slashes (5 posiions): / / / / (1) The firs posiion is for he arrival process: M: exponenial (M here sands for Markovian arrivals); E k : Erlang of order k; H k : hyperexponenial of order k; D: consan; PH: phase ype; GI: general uncorrelaed; G: general correlaed. Lecure: Basic definiions and relaions 5

6 Some non-renewal well-known coninuous-ime processes: IPP: inerruped Poisson process; SPP: swiched Poison process; MMPP: Markov modulaed Poisson process; MAP: Markovian arrival process; BMAP: Bach Markovian arrival process; Some non-renewal well-known discree-ime processes: IBP: inerruped Bernoulli Process; SBP: swiched Bernoulli process; MMBP: Markov modulaed Bernoulli process; D-MAP: discree-ime Markovian arrival process; D-BMAP: discree-ime Bach Markovian arrival process; Lecure: Basic definiions and relaions 6

7 Second posiion: service process: M: exponenial; E k : Erlang of order k; H k : hyperexponenial of order k; D: consan; PH: phase ype; G: general. Noe he following: service imes are usually assumed o be uncorrelaed (here G implies uncorrelaed); someimes his may no be rue. Third place: number of servers; here mus be a leas one; can be. Lecure: Basic definiions and relaions 7

8 Some examples of queuing sysems in Kendall s noaion are as follows: M/M/1 M/PH/10 MAP/PH/1. (2) oher places (4h and 5h) are assumed o be infinie. Fourh place: capaciy of he sysem: should be a leas 1; capaciy of no he number of waiing posiions; capaciy = # of waiing posiions # of servers. M/PH/10/N, M/M/1/K. (3) if his place may be omied; MAP/PH/1. (4) Lecure: Basic definiions and relaions 8

9 Fifh place: arrival populaion: can be finie: arrival process of broken machines o repairmen queue (number of machines may be finie). can be infinie: call arrival process o a elephone exchange. is omied when infinie! The following is imporan: fifh posiion someimes incorrecly used o denoe he service discipline; service discipline should be given separaely (no place in Kendall noaion)! The following service disciplines are he mos common: firs come, firs served (FCFS), i.e. in order of arrivals; random order (RANDOM); las come, firs served (LCFS); processor sharing (PS). Lecure: Basic definiions and relaions 9

10 M/M/1 exponenial arrivals exponenial service Erlang arrivals E k /H k /1/K hyperexponenial service infinie # of waiing posiions limied # of waiing posiions general arrivals GI/G/1 general service general arrivals GI/M/K exponenial service K servers infinie # of waiing posiions infinie # of waiing posiions Figure 2: Differen ypes of queuing sysems and heir Kendall s noaion. Lecure: Basic definiions and relaions 10

11 2. Lile s resul Lile s resul: mos general and mos powerful resul; holds for a wide variey of queuing sysem; relaes he following: mean number of cusomers in he sysem (mean queue lengh); mean sojourn ime of cusomer in he sysem; mean number of cusomers enering he sysem per ime uni. Le us define: L: mean queue lengh; W : mean waiing ime of cusomers in he sysem (sojourn ime); λ: mean number of cusomers enering he sysem per ime uni. Lecure: Basic definiions and relaions 11

12 Lile s law: L = λw. (5) his resuls was firsly published by Lile in Noes on Lile s resul: in order o hold queue mus be sable (service rae > arrival rae); he only resul ha holds for almos all queuing sysems; λ is he acual arrival rae! rejecion due o queue is full p (1 p) Figure 3: Acual arrival rae when number of waiing posiions is limied. Lecure: Basic definiions and relaions 12

13 2.1. Inuiive explanaion of he Lile s resul Consider a queuing sysem: i does no maer wha is he arrival process; i does no maer wha is he service process; for simpliciy we assume infinie number of waiing posiions. Operaion sraegy is as follows: cusomers ener he sysem a a random ime; wai o ge service in he buffer; afer service hey leave he sysem. We consider: arrival process as cumulaive number of arrivals in [0, ); deparure process as cumulaive number of deparures in [0, ). Lecure: Basic definiions and relaions 13

14 Cumulaive number of arrivals/deparures arrival process waiing ime in he sysem if FCFS deparure process number of cusomers in he sysem Time Figure 4: Illusraion of differen quaniies associaed wih a queue. Lecure: Basic definiions and relaions 14

15 Consider a ime period T and inroduce he following noaions: N(T ): number of arrivals in he period T ; A(T ): he oal service imes of all cusomers in he period T : he area beween curves; physical meaning: he carried raffic volume. λ(t ) = N(T )/T : he average arrival rae in he period T ; W (T ) = A(T )/N(T ): mean holding ime in sysem per cusomer in period T ; L(T ) = A(T )/T : he mean number of cusomers in he sysem in he period T. Then, he following relaion beween hese variables holds: L(T ) = A(T ) T = W (T )N(T ) T = λ(t )W (T ). (6) Lecure: Basic definiions and relaions 15

16 Assume he following limiing values exis: lim λ(t ) = λ, lim T W (T ) = W. (7) T Then, he following limi also exiss: Finally, we have Lile s formula: L: mean number of cusomers; λ: rae a which cusomers ener he queue; W : mean waiing ime. The following is imporan: Lile s resul binds inpu parameer λ: we usually know; oupu parameers L and W : we do no know. lim L(T ) = L, (8) T L = λw. (9) Lecure: Basic definiions and relaions 16

17 2.2. Example: how we usually use Lile s resul Consider a simple queuing sysem, M/M/1 queuing sysem. Such queuing sysem is characerized by: renewal arrival process wih exponenially disribued inerarrival imes; exponenially disribued service imes; single server; infinie number of waiing posiions. Le us define hew following: λ: he mean arrival rae of cusomers o he sysem; µ mean service rae of he server; E[W ] mean waiing ime, E[T ] mean sojourn ime. Lecure: Basic definiions and relaions 17

18 For sabiliy we assume ρ = λ/µ < 1: in his case number of cusomers does no grow o infiniy. I can be shown ha: E[N] = we will derive his resuls laer in he following lecure. ρ (1 ρ). (10) We can ge boh mean sojourn ime and mean waiing ime: E[T ] = E[N] λ E[W ] = E[N] λ = 1 µ(1 ρ), 1 µ = ρ µ(1 ρ). (11) Noe: E[N] and E[W ] when ρ and Lile s law does no applicable! Lecure: Basic definiions and relaions 18

19 2.3. Prove of he Lile s law Noe he following: here are a number of proves for Lile s law; for specific queuing sysems hey sill appear! We give he proof for he case: ρ = λ/µ << 1; noe: we explicily imply ha sysem empies infiniely ofen. We assume he following: we sar wih empy sysem; consider ime inerval [0, ); is he ime when he sysem is also empy; A() number of arrivals in [0, ); C() number of service compleions in [0, ). Lecure: Basic definiions and relaions 19

20 Cumulaive numbers 5 4 T T 3 T T 2 T 1 a 1 a 2 a 3 a 4 a 5 d 1 d 2 d 3 d 4 d 5 Figure 5: Illusraion of imes spen in he sysem T i, i = 1, 2,.... Lecure: Basic definiions and relaions 20

21 For any ime : jobs compleed by T i 0 N(s)d(s) jobs arrived by T i. (12) Dividing by we ge: jobs compleed by T i 0 N(s)d(s) jobs arrived by T i. (13) Muliplying and dividing by C() and A(): jobs compleed by T i C() N(s)d(s) 0 C() jobs arrived by T i A() A(). (14) Taking limis as : jobs compleed by T i C() lim lim C() lim 0 N(s)d(s) lim jobs arrived by T i A() lim A(). (15) Lecure: Basic definiions and relaions 21

22 Consider: lim jobs compleed by T i C() lim C() lim 0 N(s)d(s) lim jobs arrived by T i A() lim A(). (16) We ge: E[T ]ν E[N] λe[t ], (17) ν is he mean deparure rae; λ is he mean arrival rae; E[T ] is he mean sojourn ime; E[N] is he mean number of cusomers. For sable sysems (µ > λ) we have ν = λ leading o: E[N] = λe[t ]. (18) Lecure: Basic definiions and relaions 22

23 2.4. Example Simple example: a bank visied by 24 cusomers per hour on average; cusomers form a queue when all ellers are busy; he mean queue lengh is observed o be 2.4 cusomers; he mean service ime is 5 minues; wha are he mean arrival rae, waiing ime, sojourn ime, offered raffic load? Soluion: he mean arrival rae o he sysem is: λ = he mean waiing ime of cusomers: = 0.4, cusomers per minue. (19) E[W ] = E[N] λ = = 6, minues. (20) Lecure: Basic definiions and relaions 23

24 he mean sojourn ime (ime spen by cusomer in he bank) is: E[T ] = E[W ] + 1 µ = = 11, minues. (21) he mean number of cusomers in he bank is: he offered raffic load is: E[N] = λe[t ] = = 4.4, cusomers. (22) ρ = λ 1 µ = = 2, (23) he mean number of cusomers being served a any ime is 2. Noe he following: one eller can carry load ρ < 1; if here are less han 2 ellers, queue will be unsable: number of waiing cusomers will grow o infiniy; queue is sable if ρ > λn/µ, where n is he number of ellers. Lecure: Basic definiions and relaions 24