St eady -St at e Populasi Tak -Ter bat as M od el M ar kovian

6.7 St eady -St at e Populasi Tak -Ter bat as M od el M ar kovian

² A rrival: Poisson process wit h l arrivals. ² A ssumpt ions:

– Arrivals occur one at a t ime wit h FIFO discipline. – The dist ribut ion of t he # s of arrivals bet ween t and t + s depends

only on t he lengt h of t he int erval s and not on t he st art ing point t.

– The # s of arrivals during nonoverlapping t ime int ervals are in- dependent random variables. Or fut ure arrivals occur complet ely at random, independent of t he # s of arrivals in t he past t ime int ervals.

– It has been shown t hat if int erarrival t imes ar e exponent ially & independent ly dist ribut ed, t hen t he # s of arrivals is a Poisson process. (see sect ion 5.5)

² These models are called Markovian models because of t he exponent ial dist ribut ion assumpt ions

²P n (t) : pr obability of n cust omers in syst em at t ime t . ²P n : st eady-st at e probabilit y of having n cust omers in syst em ² A queuing syst em is in st eady st at e if t he syst em in a given st at e is

independent of t ime t , i.e., P n (t) = P n : ² Sect ions deal wit h mat h models t o get a rough guide of t he syst em

behaviors, model paramet er, e.g. L , inst ead of simulat ion models which delivers a st at ist ical est imat e (e.g. b L) of t he paramet er hat .

² Two propert ies are import ant t o consider for st eady st at e: st art ing st at e and remaining in st eady st at e once it is reached.

6.7.1 M =G=1

(when N and K are in…nit e, t hey may be dropped from not at ion) ² A ssumpt ions: – mean service t imes ¹ ¡1 and variance ¾ 2 , t erdapat one server

– ½= 1=¹ < 1 dan M =G=1has a st eady-st at e probabilit y dist ribu- t ion.

– St eady-st at e charact erist ics: Table 6.3.

– What is P 0 ? – What is 1 ¡ P 0 ?

– What is L ¡ L Q ?

A ssumpt ions: – exponent ially dist ribut ed

– mean: 1/ ¹ = st andar deviasi dari dist ribusi ekponesial – variance: 1/ ¹ 2 – st eady-st at e paramet ers shown in Table 6.4. – e¤ect of ½, and L and w

¤ examples 7.11 & 7.12 e¤ect of ut ilizat ion & ser vice var iabilit y – coe¢ cient of variat ion (cv) :

¤ (cv) 2 = V (x)=[E(x)] 2

¤ Figure 7.12 Relat ionship bet ween M / G / 1 and M / M / 1, and

M / G / c and M / M / c Correct ion fact or for L Q and w Q : Rewrit e L Q and w Q for M/ G/ 1 queue (Table 7.3) in t erms of t he coe¢ -

cient of variat ion and compare it wit h t hat for M/ M/ 1 (Table 7.4). Correc- t ion fact or is (1 + (cv) 2 )=2, see equat ion (7.19) . The same corr ect ion fact or can also be applied t o M/ G/ c and M/ M/ c t o obt ain L Q and w Q .