t } = Number of calls in progress at time t. Engsett Model (Erlang-B)

Transcription

1 Engsett Model (Erlang-B) A B Desrpton: Bloed-alls lost model Consder a entral exhange wth users (susrers) sharng truns (truns). When >, long ours. Ths s the ase of prnpal nterest. Assume that the truns are for long-dstane alls to other exhanges, so none of the users speas over these truns to any other of the users. Idle users eah generate/ntate alls at rate ndependent, exponental. dle users plaes next all attempt after E ( ) tme passes dle users atvate E ( ) Busy users spea for E ( ) duratons, ndependently. Busy users deatvate E ( ) Atve users eah termnate alls at rate Bloed alls alls arrvng when all truns are usy. Bloed alls are lost. Users wll not try to generate new all attempts mmedately when loed (.e., no retral/redal). Rather, suh an unserved user smply returns to the pool of dle users who generate new requests for serve at a omned rate of ( ). Let { X() t } Numer of alls n progress at tme t. Numer of usy truns at tme t. X() t () X t numer of dle users/susrers at tme t. Under our assumptons, { X () t } s a tme-ontnuous homogeneous Marov han. Q matrx dervaton: Suppose there are atve users at tme t. P all of these are stll atve at tme t+ ( )

2 P( st one s stll atve) P( 2 nd one s stll atve ) P( th one s stll atve) ( e ) e ( At least one hangs up etween t and ) ( ) P ( one ( or more ) hangs up etween t and ) P e lm Therefore, Q, Smlar reasonng gves ( ) Q, + (-) (-2) (-+) (-) (-+) (+) - (-+) (-) + (-) (+) ( ) ( ( ) + ( + ) ) ( + ) ( + ( ) ) ( ) + ( + ) ( ( + ) + ( ) ) ( ) Underlyng fat: If T, T 2,, T n are ndependent exponental random varale wth T ~ E ( α ), then, to frst order n, the proalty that the frst of these exponental los to go off does so n [, ] s ( α+ α2 + + αn ). [ ] [ T n] st Pr one s efore Pr At least one efore Pr >, n ( ) n n α α PT > e e n α + o + o Dynam equatons: p () t p() t Q() t where ( ) Pr ( ) n ( ) α ( ) p t X t. Ths s

3 () () ( ) dp t dp t dp t,,,, Or, equvalently, () dp t t p t p t ( p (),, (),, () ) ( + ) ( + ( ) ) ( + ) ( ) p ( t) ( ( ) ) p() t ( ) p+ () t ; - d p t p t q t p t q s or Easer to get from ths from () () () () ( ),, Instantaneous rate of Instantaneous flow of Instantaneous flow of hange of proalty, and proalty nto state proalty out of state of state the dagram. (-+) (-) (-+) (+) Ether way, we have: dp () t p() t p() t dp () t ( + ) p ( t) ( + ( ) ) p() t + ( + ) p+ () t dp () t ( + ) p ( t) p( t) Equlrum dstruton: Trunated nomal: p, : () dp t Set,, Ths wll gve

4 p p ( + ( ) ) p ( + ) p + ( + ) p+ p ( + ) p Use - equatons of these + equatons plus From parttonng, we get p. (-) (-+) + - ( ) ( ) + + (+) p + p p p p ; ( ( ) )( ( 2) ) ( ) ( ( ) )( ( 2) ) ( ) p ( ) p p p 2 2! ( )!! p p p!! ( )! p p p p ; For small values of, ths dstruton s slanted heavly toward the small values of. It s unmodal for ntermedate values of, and t s slated heavly toward the large values of for large values of.

5 Engsett wth, 75 rho.5 rho.5 rho rho Def: P Pr[ all attempt s loed] Pr[ get a usy sgnal] Pr[ long-term fraton of all the all attempts that get loed] P ( ) ( ) Proof. Sne the Engsett arrval proess s state-dependent and hene not Posson, Engsett arrvals do not see steady-state onons, so the long proalty s not equal to p. We may assume that steady-state onons preval; they wll n the long run regardless of the value of eause we have an rredule, fnte-state tmeontnuous han. In the steady state, the value of p represents the fraton of the tme axs durng whh the system s n state, or equvalently the proalty that the system s n state at a randomly hosen nstant. However, the densty of allng attempts n the Engsett model vares wth the state of the system. When n state, all attempts our at rate (-). The fraton of all all attempts that our when the system s n state s not ( ) p p ut rather f ( ) p. p p ( ) ( ) Sne all attempts get loed f and only f they our when the system s n state, the long proalty s

6 P f ( ) p ( ) p lm, lm, lm P P P p p ( ) ( ) Proof. Sne the term has the hghest order n ( ) Proof. ( ), ( ) ( ) and ( ) ( ) lm ( ) ( ) ( ) lm P lm ( ) Proof. For < <! ( ) ( )! ( )!! ( )!!!( )!! ( )! ( ) ( + ) ( ) ( ( ) ) ( ( ) ) ( ) ( + ) ( ( + ) ) ( ( ) )( ) + ( )

7 Hene, lm P ( ) lm ( ) ( ) lm ( ) P for. If there s only one fewer trun than susrer ( -), then Proof. Note that ( + ) P +. Hene, ( ) ( ) ( ) Beause when, ( ) ( ) ( ), we have ( ) ( ) ( + ). + (( + ) ) ( + ) P + (( ) ) ( + )( + ) + + P < p for < Proof. We want to ompare P ( ) ( ) and p

8 ( ) < ( ) ( ) < ( ) ( ) ( ) ( ) ( ) < < P < p Cauton! P Pr[X(t) and next p-up preedes next hang-up] Pr[X(t) and next p-up preedes next hang-up] P Pr[U < V n state ] U ~ V ~ E, and U V where E (( ) ), ( ) ( ) ( ) p ( ) + ( ) + p Erlang Model Defnton Infnte populaton of users ntates alls at a omned Posson rate regardless of how many alls ( ) are n progress Agan Bloed alls lost. Holdng tmes are..d. E ( ) B State Dagram:

9 - + - (+) p p p p p + ( + ) p+ ( + ) p ; p p p p!! (Trunated Posson). Proof. Vertal dashed lne gve + - (+) p p p p p p ( + ) p p p p Hene, p p.! From p, p p! Thus, p Proof 2.!!! trunated Posson

10 Use formula p mples R.! P! p! Proof. P R R p f p, where R, R rr r, r p p p p Proof 2. Beause, we already now that P f p.. Ths