Basic Queueing Theory M/M/-/- Type Queues

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Clemence Della Sanders
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1 Basc uug Thoy M/M/-/- Ty uus Coygh 22, Sajay K. Bos Kdall s Noao o uus A/B/C/D/ Shohad oao wh A, B, C, D, dscb h quu Alcabl o a lag ub o sl quug scaos Coygh 22, Sajay K. Bos 2

2 Kdall s Noao o uus A/B/C/D/ A B C D I-aval dsbuo Svc dsbuo Nub o svs Mau ub o jobs ha ca b h h sys wag ad svc Daul o ub o wag osos uug Dscl FCFS, LCFS, SIRO c. Daul s FCFS M oal D dsc laga od G gal M/M/ o M/M// Sgl sv quu wh osso avals, oally dsbud svc s ad ub o wag osos Coygh 22, Sajay K. Bos 3 Ll s Rsul NW 2.9 N q W q 2. Rsul holds gal o vually all ys o quug suaos wh Ma aval a o jobs ha acually h sys Jobs blocd ad usd y o h sys wll o b coud Coygh 22, Sajay K. Bos 4 2

Daul s FCFS M oal D dsc laga od G gal M/M/ o M/M// Sgl sv quu wh osso avals, oally dsbud svc s ad ub o wag osos Coygh

3 Ll s Rsul Nub o cusos avals/daus N α, Avals, β, Daus, cs by o T Gahcal Illusao/Vcao o Ll's Rsul Coygh 22, Sajay K. Bos 5 Ll s Rsul Cosd h val, wh s lag,.. Aa aa bw α ad β a [ α β ] d Avag T W s sys l Aa α Avag Nub N sys l Aa l α Aa α Sc, l α Tho, N W Coygh 22, Sajay K. Bos 6 3

4 Th ASTA oy osso Avals S T Avags sys s sa a q a aval a ds h sys sa N b h acual ub h sys a A, b h v o a aval h val, Th q l l N A, A, N N A, bcaus A, N A, Coygh 22, Sajay K. Bos 7 qulbu Soluos o M/M/-/- uus Mhod : Oba h dal-dc quaos as Sco.2 o Sco 2.2. Solv hs ud qulbu codos alog wh h oalzao codo. Mhod 2: Dcly w h low balac quaos o o choc o closd boudas as llusad Sco 2.2 ad solv hs alog wh h oalzao codo. Mhod 3: Idy h aas o h bh-dah Maov cha o h quu ad dcly us quaos 2.7 ad 2.8 as gv Sco 2.2. I h ollowg, w hav usd hs aoach Coygh 22, Sajay K. Bos 8 4

Mhod 2: Dcly w h low balac quaos o o choc o closd boudas as llusad Sco 2.2 ad solv hs alog wh h oalzao codo.

5 M/M/ o M/M// uu,2,3,... Fo < N W N Usg Ll s Rsul 2 W q W N q Wq Usg Ll s Rsul Coygh 22, Sajay K. Bos 9 M/M// uu wh Dscouagd Avals,2,3,... Fo / <! N W N 2 Ll s Rsul cv Aval Ra Coygh 22, Sajay K. Bos 5

6 6 Coygh 22, Sajay K. Bos M/M// uu wh Dscouagd Avals I hs cas, ASTA s o alcabl as h ovall aval ocss s o osso π avg cuso ss sys bo jog h sys b h v o a aval, s h v o h sys bg sa π! / π / /! / 2 W π Coygh 22, Sajay K. Bos 2 M/M// uu svs, ub o wag osos Fo / < o o >!!!! C!, 2.8 quug lag s C-Foula

7 M/M// uu sv loss sys, o wag < ohws Blocg o Loss Codo Fo < ohws o! ohws! Coygh 22, Sajay K. Bos 3 M/M// uu sv loss sys, o wag Sl odl o a lho chag wh a l s gv oly o s avalabl; ohws h call s los Blocg obably B, a aval ds all svs busy ad lavs whou svc B,! lag s B-Foula 2.2 B, B, B, 2.22 B, Coygh 22, Sajay K. Bos 4 7

8 M/M//K uu sgl sv quu wh K- wag osos < K ohws Blocg o Loss Codo K ohws Fo < K o K ohws Coygh 22, Sajay K. Bos 5 M/M//-/K uu sgl sv, ub o wag osos, cuso oulao K K < K ohws K ohws Blocg o Loss Codo Fo K! K!,..,K 2.25 < K K K!! 2.26 Coygh 22, Sajay K. Bos 6 8

9 9 Coygh 22, Sajay K. Bos 7 Dlay Aalyss o a FCFS M/M// uu Sco 2.6. : uug Dlay o coug svc o a aval d, cd F, L s LT W: Toal Dlay wag ad svc o a aval d W, cd F W, L W s LT W WT T: Svc T T T F s s L T Kowg h dsbuo o h W o, h dsbuo o h oh ay b oud Sc T ] *[ W W s L s s L 2.3 Coygh 22, Sajay K. Bos 8 Fo a acula aval o s - F quug dlay quug [Σ quug aval oud jobs sys]!! d d d F lag- dsbuo o su o oal.v.s 2.3 δ d df 2.32 W d

WT T: Svc T T T F s s L T Kowg h dsbuo o h W o, h dsbuo o h oh ay b oud Sc T ] *[ W W s L s s L 2.

10 Aval o cuso o s T s sys by h cuso o s Avals cog whl h cuso o s s h sys Dau o cuso o s ASTA alcabl o hs quu N ad N s by a aval sa as h -avagd valus Aval/Dau o Cuso/Job o Is o a FCFS M/M/uu L N * Nub h sys ha a job wll s l bhd wh das *N * o N *,,..., Coygh 22, Sajay K. Bos 9 T s sys by h cuso o s Dau o cuso o s Fo a FCFS quu, ub l bhd by a job wll b qual o h ub avg whl s h sys. Aval o cuso o s * G z z Avals cog whl h cuso o s s h sys * z W * dg z N dz z d L! W W z dlw s W ds * z s d Coygh 22, Sajay K. Bos 2

11 Coygh 22, Sajay K. Bos 2 A oa gal obsvao ca also b ad alog h ls o q Cosd h ub avg o a osso ocss wh a a ado val T wh L T slt T. Th gag uco Gz o hs wll b gv by z L z G T ad h a ub wll b N T Ths sul wll b oud o b usul vaous lacs ou subsqu aalyss. Coygh 22, Sajay K. Bos 22 Dlay Aalyss o h FCFS M/M// uu Sco Usg a aoach sla o ha usd o h M/M/ quu, w oba h ollowg!! u δ! ] [! W S Sco o h dals ad h da ss

Th gag uco Gz o hs wll b gv by z L z G T ad h a ub wll b N T Ths sul wll b oud o b usul vaous lacs ou

The effect on the Asian option price times between the averaging. Mark Ioffe

The effect on the Asian option price times between the averaging. Mark Ioffe 866 U Naos Plaza u 566 Nw Yok NY 7 Pho: 3 355 Fa: 4 668 fo@gach.co www.gach.co h ffc o h sa opo pc s bw h avagg Mak Ioff bsac h acl s o h calculao of h pc of sa opo. I pacula w aalz h ffc o h opo pc s