Transient Analysis of Two-dimensional State M/G/1 Queueing Model with Multiple Vacations and Bernoulli Schedule

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1 Inrnaonal Journal of Compur Applcaons ( ) Volum 4 No.3, Fbruary 22 Transn Analyss of Two-dmnsonal Sa M/G/ Quung Modl wh Mulpl Vacaons and Brnoull Schdul Indra Assoca rofssor Dparmn of Sascs and Opraonal Rsarch, Kurukshra Unvrsy Kurukshra, Haryana, 369, INDIA Rnu Unvrsy Rsarch Scholar Dparmn of Sascs and Opraonal Rsarch, Kurukshra Unvrsy Kurukshra, Haryana, 369, INDIA ABSTRACT Ths papr s concrnd wh h ransn analyss of wodmnsonal M/G/ quung modl wh gnral vacaon m basd on Brnoull schdul undr mulpl vacaon polcy. As soon as a srvc gs compld, h srvr may ak a vacaon or may connu sayng n h sysm. Whnvr no cusomrs ar prsn, afr a srvc complon or a vacaon complon, h srvr always aks a vacaon. Laplac ransforms of probabls of ac numbr of arrvals & dparurs by a gvn m and numbr of uns arrv by m usng supplmnary varabl chnqu ar oband. Th mphass n hs papr s horcal bu numrcal assssmn of opraonal consquncs s also gvn and prsnd graphcally. Fnally, som spcal cass of nrs ar drvd hr from. Ky words: Two-dmnsonal quung modl; Mulpl Vacaon; Brnoull Schdul; Non-Markovan quu; Supplmnary varabl chnqu.. INTRODUCTION In mos of h quung modls, on complon of srvc o h sng cusomrs, h srvr says n h mpy sysm wang for a nw arrval. Bu hr ar suaons whr f h srvr afr complng h srvc of a cusomr fnds h quu mpy, gos away for a lngh of m calld vacaon. Ths m may b ulzd by h srvr o carry ou som addonal work. On rurn from a vacaon f fnds on or mor cusomrs wang, aks hm for srvc on a on-by-on bass unl h sysm mps, afr whch m aks anohr vacaon. Howvr, f, on rurn from a vacaon, fnds no cusomr wang, hn, n h cas of sngl vacaon, rmans dorman unl a las on cusomr arrvs, whras n h cas of mulpl vacaon mmdaly procds for anohr vacaon and connus n hs mannr unl fnds a las on wang cusomr upon rurn from a vacaon. Usfulnss of vacaon modls n quung hory hav bn wll sablshd as consdrd bng an ffcv nsrumn n modllng and analyss of communcaon nworks, manufacurng and producon sysms. Durng h las hr dcads, quung sysms wh vacaons hav bn sudd nsvly. Concrnng quung modls wh srvr vacaons, Dosh [3] provds clln survy on vacaon modls. Takag [4] also prsns h analyss of a vary of quung sysms wh vacaons. Rcnly, Tan and Zhang [9,], provd an updad and comprhnsv ramn of varous vacaon quung sysms. Th quung modls of smlar naur hav also bn rpord by a numbr of auhors [,2,8]. Alhough h sng rsuls of vacaon quus rpord n lraur ar oband by dffrn mhods, bu n hs papr, basd on Brnoull schdul w consdr a sngl srvr wo-dmnsonal M/G/ vacaon quu nvolvng only fn sums for h probably ha acly arrvals and j srvcs occur ovr a m nrval of lngh n a quung sysm ha s dl a h bgnnng of h nrval. Brnoull vacaon schdul was proposd by Klson and Srv [6]. I s characrzd by h faur ha f h quu s mpy afr a srvc complon hn h srvr bcoms nacv and bgns a vacaon prod. If h quu s no mpy, hn wh probably p h srvr may say n h sysm provdng srvc, or wh probably (-p) h may ak a vacaon of random lngh. D() η() η(u)du Th advanag of h Brnoull Schdul s h snc of a conrol paramr p. And s shown ha h ransn sa probabls can b asly compud wh rcurrnc rlaons. Varous aspcs of Brnoull vacaon modls for sngl srvr quung sysms hav bn sudd by Srv[7], Ramaswamy and Srv[3], Dosh[4], and Coopr[2] among svral ohrs. Th rmandr of hs papr s organzd as follows. In h n scon, w provd a rlav formal dscrpon of h quung modl and som noaons. Scon 3 gvs dfnon and dffrnc-dffrnal quaons of wodmnsonal quung modl. In scon 4, w nvsga h ransn-sa soluon for h quung modl by usng h supplmnary varabl chnqu. Scon 5 dals wh h vrfcaon of modl and h prformanc masurs and numrcal rsuls wh hr graphcal rprsnaon ar gvn whn h srvc m s ponnal. In scon 6, w dscuss som spcal cass. Conclusons ar drawn n scon MODEL DESCRITION AND NOTATIONS In hs papr, w consdr M/G/ quung modl undr followng spcfcaon. I s assumd ha cusomrs arrv accordng o a posson procss wh ra. Arrvng cusomrs form a sngl wang ln basd on h ordr of hr arrvals; ha s, hy ar quud accordng o h frscom-frs-srvd (FCFS) dscpln. Th srvc m s assumd o b gnrally dsrbud wh probably dnsy funcon D() and η()δ s h frs ordr 7

2 Inrnaonal Journal of Compur Applcaons ( ) Volum 4 No.3, Fbruary 22 probably ha h corrspondng srvc m wll b compld n m (, Δ) provdd h sam had no bn compld ll m. And Cusomrs connu o arrv accordng o a posson procss. W also assum ha h srvr aks a random vacaon ach m h sysm s mpy. Whn h srvr rurns from a vacaon and fnds a las on cusomr wang n h sysm, h bgns h srvc mmdaly unl h sysm s mpy agan, ohrws, anohr vacaon s akn and h vacaon ms ar ponnally dsrbud wh paramr w. If h quu s no mpy, hn wh probably p h srvr may say n h sysm provdng srvc, or wh probably (-p) h may ak a vacaon of random lngh. W assum ha all h consdrd varabls ar muually ndpndn. s f (s) f d R(s) (2.) and δ,j N a,b n,n2 ; whn j ; whn j (s) (s a) (s b) Th Laplac nvrs of n n2 (2.2) n mk mk ak Q(p) d Q(p) mk s (p a k ) (p) k (m k )!( )! dp (p) pak a a k for k Whr, m m2 m (p) (p a n ) (p a 2) (p a n ) Q(p) s polynomal of dgr m m m m N a, b n,n n m δm, m2 (n g) n m g n 2 n 2 m a ( ) m (n 2 m)!(m )!(b a) n n m b m δ ( ) m2 (n n 2 g) m 2 m (n m)!(m )!(a b) g (2.3) 3. THE TWO-DIMENSIONAL STATE MODEL In conras o h classcal dvlopmn, w bas our analyss of h M/G/ quu on a modl n whch h sa of h sysm s gvn by (, j), whr s h numbr of arrvals and j s h numbr of dparurs unl m. W dno h sa probabls for h modl as j m,, j. Th soluon for, provds consdrabl nformaon concrnng h ransn bhavour of h quung modl. W hav, j, B (, )Δ Th probably ha hr ar acly arrvals and j dparurs by m and h srvr s busy n rlaon o h quu and h lapsd srvc m ls bwn and Δ, j<, j, V = Th probably ha hr ar acly arrvals and j dparurs by m and h srvr s on vacaon, j, j = Th probably ha hr ar acly arrvals and j dparurs by m. j Th soluon for, j provds consdrabl nformaon concrnng h ransn bhavour of h quu. Th dffrnc-dffrnal quaons govrnng h sysm ar,j,b(,),j,b(,) η(),j,b(,) (,)(-δ ) -,j,b ; j (3.),j,V =-(+w),j,v + -,j,v + j,,j-,b -,j (-δ )(-p) η() (,)d ; j ; (3.2) η() (,) d (-δ ),,V,,V,,B, ; (3.3) Th appropra boundary condons ar (,)Δ w (-δ )p η() (,)d,j,b,j,v j,,j,b ; j (3.4) Clarly,,j=,j,V +,j,b(-δ,j) ; j (3.5) Inally, h sysm sars whn hr ar no uns n h sysm and h srvr s on vacaon,..,, V (),,, B (,) 4. SOLUTION OF THE ROBLEM Usng Laplac ransforms n abov quaons (3.) o (3.4) along wh nal condons and solvng rcursvly, w hav,, V(s) s (4.),,V (s)= (s++w) (s+) (4.2) +w,,,b(s)= wn -k+,(s) k= k- (s++η(u))du (k-)! d (4.3) -j -j-m -δ +w, m+,,j,v (s) (-p) N (s) (-j -m+-δ m+, ),(δ m+, ) m= η() j+m, j-, B (,s) d 8

3 Inrnaonal Journal of Compur Applcaons ( ) Volum 4 No.3, Fbruary 22 -h h -k,j,b(s)= w p h= k=j+ k-j k-j-m -δ +w, m, (-p) N -k+,δ (s) m, m= ; j (4.4) (-k)h+(k-j-m-δ m, )(-h) - (s++η(u))du ((-k)h+(k-j-m-δ m,)(-h))! d η() (m+j)(-h)+kh,j-,b (,s)d ; >j> (4.5) Takng h Laplac nvrs ransforms of h abov quaons (3.6) o (3.), w hav,, V (4.6) ; > (4.7) δ,h k w,,v k w k k! w +w,,,b= w N -k+, k=,j,v k- (+η(u))du (k-)! -j m= -j-m -δm+, +w, j+m, j-, B ;> (4.8) (-p) N η() ; j (4.9),j,B (,) d -h h -k = w p h= k=j+ (-j -m+-δ m+, ), (δ m+, ) k-j k-j-m -δm+, +w, (-p) N -k+,δ m+, m= δ,h (-k)h+(k-j-m-δ m, )(-h) - (+η(u))du ((-k)h+(k-j-m-δ η() ; >j> (4.) m, (m+j)(-h)+kh,j-,b )(-h))! (,)d 5. ERFORMANCES MEASURES OF THE SYSTEM 5. Th Laplac ransform (s) of h probably ha acly uns arrv by m s; (s) (s) (s)( δ ) (s),j,v,j,b,j,j j j (s ) ; > (5.) And s Invrs Laplac ransform s ( ), j j! Th Laplac ransform of h man numbr of h arrvals s, (s) 2 s Th arrvals follow a osson dsrbuon as h probably of h oal numbr of arrvals s no affcd by h vacaon ms and brakdowns of h srvr. Hnc, a vrfcaon And h numrcal rsuls for h probabls of ac numbr of arrvals whn h srvr s busy.., j, B, whn h srvr s on vacaon.. j=, j, V,ar compud for dffrn ss of j= paramr and ar summarzd n Tabl. Tabl- s basd on h rlaonshp (5.) and s las column shows compl agrmn wh h Tabl of gdn and Rosnshn []. 5.2 Th numrcal rsuls for h probabls ha acly j numbr of cusomrs hav bn srvd whn h srvr s on vacaon.., j, V, whn h srvr s busy.. =j =j j j, j, B,j,V (s),j,b(s)( δ,j) s ( δ ),j,v,j,b,j ar compud for dffrn ss of paramrs ( 2, μ 3, w=2, =2, p=.4,.6,.8) and ar basd on h rlaonshp j=, j whr, j s =j dfnd n quaon (3.5). By adjusng h valu of p, w can conrol h congson of h sysm. And from h numrcal rsuls s obvous ha as p ncrass h probably of dparurs ncrass whn h srvr s busy. In fgs , h graphcal rprsnaon of j wh h varaon of p has bn shown. 9

4 Ulzaon m of h srvr robably of n Cusomrs robably of dparurs whn srvr s busy robably of n Cusomrs whn srvr s busy robably of dparurs whn srvr s on vacaon robably of n Cusomrs whn srvr s on vacon Inrnaonal Journal of Compur Applcaons ( ) Volum 4 No.3, Fbruary p=.4 p=.6 p= = =2 =3 =4 = Numbr of dparurs(j).6.2 fg. 5. p=.4 p=.6 p= Numbr of Cusomrs(n) Fg. 5.3 = =2 =3 =4 = Numbr of Cusomrs (n) Fg Numbr of dparurs (j) fg Th probably of acly n cusomrs n h sysm a m, dnod by of ()., j = n, B j+n, j, B j= () can b prssd n rms n Cusomrs whn h srvr s busy,.. Cusomrs whn h srvr s on vacaon,. = n, V j+n, j, V j= And ar basd on h = rlaonshp n j+n, j j= whr n= n, V+ n, B and n, V, n, B an d n ar compud for dffrn valus of paramrs ( 2, μ 3, w=2, p=.4). In fgs. 5.3 o 5.5, h graphcal of n, V, n, B has bn shown. rprsnaon and n wh h varaon of m.3.2. = =2 =3 =4 = Numbr of Cusomrs (n) Fg Th srvr s ulzaon m, and h srvr s vacaon m.. h fracon of m h srvr s busy & h fracon of m h srvr s on vacaon unl m can. also b prssd n rms of, j Thus h srvr s ulzaon m s U=, j, B. And h srvr s vacaon m s = j= V=, j, V. = j= Vacaon Ra (W) Fg.5.6 2

5 Vacaon m of h srvr Inrnaonal Journal of Compur Applcaons ( ) Volum 4 No.3, Fbruary CONCLUSION Vacaon Ra(w) Fg SECIAL CASES 6. Whn hr s no Brnoull schdul,.. by subsung p= n h dffrnc dffrnal qns., hn abov dscrbd modl rducs o hausv srvc dscpln and oband rsuls concd wh rsuls on Indra [5]. 6.2 Whn h srvc m s ponnal,.. by lng η() = μ n dffrnc dffrnal quaons, w hav -j -j-m -δm, +w,,j,v (-j-m+-δ m, ), (δ m, ) m= = μ(-p) N,, B,j,B k-j m= * (m+j)(-h)+mh, j-, B - k-j-m m -δm, δ,h w N -h h -k = μw p h= k=j+ (-p) +μ,+w, (k-j-m+)(-h)+(-k+)h, -k+,δ m, ; j (6.) m, m, ; > (6.2) N (m+j)(-h)+kh,j-,b ; < j < (6.3) 6.3 Along wh h cas-6. & cas 6.2, whn h srvr s nsananously avalabl.. no dscpln of vacaon. Lng w n h dffrnc-dffrnal qns., w hav,,, V μ k j j,j,j,b= μ! K k! r m r! m! jk m! μ μ m k! -k mk μ mk m= r ; j Thn rsuls concd wh qn. () of gdn and Rosnshn []. Th Two-dmnsonal sa M/G/ quung sysm wh ponnal vacaon has bn nvsgad. Th numrcal analyss clarly dmonsras h manngful mpac of h vacaons on h sysm prformancs. By adjusng h valu of paramr p, w can conrol h congson of h sysm. And ulzaon m ncrass and dcrass wh rspc o h ncras and dcras n vacaon m. Furhr, wh h hlp of wo-dmnsonal modl, w can provd h nformaon on how h sysm has bn oprad up unl m. And w hav shown numrcally as wll as analycally ha probably of arrvals s no affcd by occurrnc of vacaons. 8. REFERENCES [] Alfa,A.S.,(23), Vacaon modl n dscr m, Quung Sysm, Vol.44 (), pp.5-3. [2] Choudhury, G. (2), An M X /G/ quung sysm wh a sup prod and a vacaon prod, Quung Sysm, Vol. 36, pp [3] Dosh,B.T.(986), Quung sysms wh vacaons-a survy, quung sys.,vol., pp [4] Dosh,B.T. (99), Sngl srvr quus wh vacaons, n: H. Takag (Ed.), Sochasc Analyss of Compur and Communcaon Sysms, Norh- Holland, Amsrdam, pp [5] Indra, Som wo-sa sngl srvr quung modls wh vacaon or las arrval run, h.d. hss (994), Kurukshra Unvrsy, Kurukshra. [6] Klson J. and Srv, L.D., Dynamcs of h M/G/ vacaon modl, Opraons Rsarch, 35(4), 987, [7] L.D. Srv, Avrag dlay appromaon of M/G/ cyclc srvc quu wh Brnoull schduls, IEEE Slcd Ara of Communcaon, 4(986), [8] Lvy, Y. and Ychal, U., (975), Ulzaon of dl m n an M/G/ quung sysm, Managmn Scnc, Vol. 22, No. 2, pp [9] NN. Tan and Z.G. Zhang. (22). Th Dscr- Tm GI/Go/ Quu wh Mulpl Vacaons. Quung Sysms, Vol. 4,pp [] N. Tan and Z.G. Zhang, Vacaon Quung Modls: Thory and Applcaons, Sprngr, Nw York 26. [] gdn, C.D. and Rosnshn, M. (982), Som nw rsuls for h M/M/ quu, Mg Sc., Vol. 28, (982). [2] R. B. Coopr, Quus srvd n Cyclc Ordr: Wang Tms, Th Bll Sysm Tch. J., 49(97), [3] R. Ramaswam, L.D. Srv, Th busy prod of h M/G/ vacaon modl wh a Brnoull schdul, Sochasc Modls, 4(988), [4] Takag, H. (99). Quung Analyss: A Foundaon of rformanc valuaon, vacaon and prory sysms, ar, Norh-Holland, Amsrdam. 2

6 Inrnaonal Journal of Compur Applcaons ( ) Volum 4 No.3, Fbruary 22 Tabl ; robably ha acly uns hav arrvd by m ( w=; p=.4, =3) ( ), j, V, j, B!, j j= j= j=

Advanced Queueing Theory. M/G/1 Queueing Systems

Advanced Queueing Theory. M/G/1 Queueing Systems Advand Quung Thory Ths slds ar rad by Dr. Yh Huang of Gorg Mason Unvrsy. Sudns rgsrd n Dr. Huang's ourss a GMU an ma a sngl mahn-radabl opy and prn a sngl opy of ah sld for hr own rfrn, so long as ah sld