Research Article On the Steady-State System Size Distribution for a Discrete-Time Geo/G/1 Repairable Queue

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1 Dscrete Dyamcs ature ad Socety, Artcle ID , 9 pages Research Artcle O the Steady-State System Sze Dstrbuto for a Dscrete-Tme Geo/G/1 Reparable Queue Reb Lu ad Zhaohu Deg School of Mathematcs ad Statstcs, Chogqg Uversty of Techology, Chogqg , Cha Correspodece should be addressed to Reb Lu; lurb888@126.com Receved 28 September 2013; Accepted 10 February 2014; Publshed 19 March 2014 Academc Edtor: Carlo Pccard Copyrght 2014 R. Lu ad Z. Deg. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. Ths paper studes a dscrete-tme -polcy Geo/G/1 queueg system wth feedback ad reparable server. Wth a probablstc aalyss method ad reewal process theory, the steady-state system sze dstrbuto s derved. Further, the steady-state system sze dstrbuto derved ths work s extremely sutable for umercal calculatos. umercal example llustratesthe mportat applcato of steady-state system sze dstrbuto system capacty desg for a etwork access proxy system. 1. Itroducto Over the past years, the aalyss of dscrete-tme queueg systems has receved more atteto the lteratures. Ths s because the dscrete-tme queues are more approprate tha ther cotuous-tme couterparts ther applcablty for may computer ad commucato systems whch tme s dvded to fxed-legth tme tervals (for detals, e.g., see [1 4]). It s well kow that the -polcy troduced by Yad ad aor [5] s very mportat the theory ad applcatos of queueg models. For detaled overvews of the cotuous-tme queues wth -polcy, the reader s referred to a excellet survey by Tadj ad Choudhury [6]. Related works o the dscrete-tme queues wth -polcy ca be foud [7, 8]. Recetly, We et al. [9] studed the traset ad equlbrum propertes of the queue legth for a - polcy Geo/G/1 queue wth varable put rate ad dcated that the stochastc decomposto property o loger holds. Wag [10] cosdered a radom -polcy Geo/G/1 queue wth start-up ad closedow tmes ad obtaed aalytc solutos of system sze, legths of state perods, ad sojour tme.asfaraskowtotheauthors,veryfewpapersabout dscrete-tme -polcy queueg systems are avalable. Feedback was troduced by Takács [11] ad sce the may papers have appeared about ths topc o cotuoustme case (see, e.g., [12 14]). Ateca ad Moreo[15] frst exteded the study to the dscrete-tme Geo X /G H /1 retral queue. The pheomeo of feedback has may practcal applcatos, for example, telecommucato systems where messages that produce errors at the destato are set aga, a call ceter where customers may call aga (repeat ther servce) f ther problems are ot completely solved after theservceadsoforth.however,relatvetocotuous-tme feedback queues, ther dscrete-tme couterparts receved very lttle atteto the lterature. A remarkable pheomeo a queueg system s ts server breakdow. For stace, computer systems, the mache may be subject to scheduled backups ad uforeseeable falures. Queues wth server breakdows ca also be called reparable queues. For oly the studes o dscrete-tme reparable queues, the reader may refer to Wag ad Zhag [16], Tag et al. [17], ad Lu ad Gao [18]. However, oly very few works the lterature cocered wth -polcy queues wth feedback ad reparable server have bee doe. Partcularly, the researches of the dscretetme -polcy queues wth feedback ad reparable server are ot foud. Further, most works of the -polcy queue, the probablty geeratg fucto (PGF) of steadystate system sze dstrbuto rather tha steady-state system sze dstrbuto was obtaed. I ths paper, we vestgate a dscrete-tme -polcy Geo/G/1 queueg system wth feedback ad reparable server. Wth a probablty aalyss

2 2 Dscrete Dyamcs ature ad Socety method ad reewal process theory, we derve the steadystatesystemszedstrbuto.itsotedthatthesteadystate system sze dstrbuto derved ths paper s qute sutable for umercal calculatos. What s more, umercal examples llustrate the applcatos of steady-state system sze dstrbuto system capacty desg for a etwork access proxy system. The rest of the paper s orgazed as follows. The ext secto presets the model assumptos ad some prelmares. I Secto3 we dscuss the traset system sze dstrbuto ad derve the steady-state system sze dstrbuto ad ts PGF. Secto 4 umercally shows the mportat applcato of system sze dstrbuto system capacty for a etwork access proxy system. Coclusos are fally draw Secto Assumptos ad Prelmares We cosder the followg dscrete-tme -polcy Geo/G/1 queueg system wth Beroull feedback ad reparable server. (1) Let the tme axs be slotted to tervals of equal legth wth the legth of a slot beg oe ut. To be more specfc, let the tme axs be marked by 0,1,...,,...Herewe dscuss the model for a late arrval system wth delayed access (LAS-DA) ad therefore a potetal arrval takes place (,)ad a potetal departure occurs (, ); for detals, see Huter [19]. The varous tme epochs volved our model ca be vewed through a self-explaatory fgure (see Fgure 1). The terarrval tmes betwee customers, τ, 1}, are depedet detcally dstrbuted (..d.) radom varables wth a geometrc dstrbuto Pτ j} p(1 p) j 1, (j 1, 0<p<1). (2) Whe there are ( 1)customers preset system the server begs to serve the frst-come, frstserved (FCFS) order utl o customer s preset. The servce tmes, χ, 1}, are..d. radom varables havg dstrbuto g j Pχ j}, j 1,wthPGFG(z) j1 g jz j, z < 1, ad mea μ. The customer whose servce has bee just fshed comes back to the queue wth probablty 1 θ(0 <θ 1) watg for the ext servce or leaves forever wth probablty θ. (3) The server may fal whe ad oly whe t s servg a customer. The faled server wll be repared mmedately. After repar, the server s as good as ew ad cotues to serve the customer whose servce has ot bee fshed yet. We assume that the servce tme for a customer s cumulatve. (4) The lfetme of server has a geometrc dstrbuto wth parameter α, adtsrepartmey obeys a arbtrary dstrbuto y j PYj}, j 1wth PGF y(z) j1 y jz j, z < 1,adamearepartmeβ. (5) All radom varables are mutually depedet. At the tal tme 0, the server begs to serve whe the umber of customers preset the system (0 )>0or the server s dle ad wats for the frst arrval whe (0 )0.Afterthe frst busy perod, the server wll take a -polcy. Remark 1. Throughout ths paper, we adopt the followg otatos. E(X) s the mea of radom varable X. P(Q) s the probablty of evet Q. ( ) deotes the system sze at epoch. C k l l!/k!(l k)!, 0 k l. p1 p. f(z) pz/(1 pz), z < 1. Defto 2. The geeralzed servce tme of a customer χ deotes the tme terval from the tme whe the server begs to serve a customer utl the servce of ths customer eds, whch cludes some possble repar tmes of the server due to ts falures the process of servg ths customer. Lemma 3 (see [17]). The PGF ad the mea of χ are gve, respectvely, by G (z) j1 P( χ j)z j G[zαy(z) z(1 α)], z <1, E( χ) d G (z) μ(1αβ), dz z1 where G(z) ad y(z) are gve by assumptos (2) ad (4). For customer,letω, χ,ad χ deote total geeralzed servce umber, total geeralzed servce tme, ad the th geeralzed servce tme before departure, respectvely. The t follows from assumpto (2) that P(Ω k) θ(1 θ) k 1, k 1,ad χ Ω 1 χ, 1, are..d. radom varables, where χ,, 1, are mutually depedet wth the same dstrbuto as χ, 1. Lemma 4. Let g j P( χ j), j 1, be the dstrbuto of total geeralzed servce tme χ for oe customer before departure. The oe gets Ω g j P G (z) j j1 1 χ j} θ(1 θ) k 1 P E( χ) d G (z) dz z1 where G(z) s gve by Lemma 3. k 1 χ j}, j 1, g j z j θ G (z), z <1, 1 (1 θ) G (z) μ(1αβ), θ Defto 5. The server busy perod deotes the tme terval from the tme whe the server begs to serve a customer utl the system becomes empty, whch cotas some possble repar tmes of server due to ts falures the process of servce. (1) (2)

3 Dscrete Dyamcs ature ad Socety 3 A A ( 1) 1 ( 1) D D potetal arrval epoch potetal departure epoch edg of repar epoch begg of repar epoch outsde observer s epoch (,(1) ) outsde observer s terval epoch after a potetal departure epoch pror to a potetal arrval Fgure 1: Varous tme epochs a late arrval system wth delayed access. Let b deote the server busy perod tated wth oe customer, ad ts PGF s b(z) j0 P(b j)zj, z < 1. Accordg to [20], the followg lemma holds. I ths case, for z < 1,thePGFsofl j,j 1ad b j,j 1are gve, respectvely, by Lemma 6. b(z) satsfes the equato b(z) G[(1 p)z pb(z)z],ad μ(1αβ), ρ < 1, E (b) θ pμ(1αβ), ρ 1, (3) P(l j k)z k f (z), j 1, f (z), j 2, P(b j k)z k b (z), j 1, b (z), j 2, (5) where ρ pμ(1αβ)/θ s the traffc testy of the cosdered queueg system. The server busy perod tated wth customers s deoted by b.the,b ca be expressed as b ζ 1 ζ, where ζ 1,...,ζ are depedet of each other ad have the same dstrbuto as b.thusthepgfofb s b (z). Defto 7. The system dle perod deotes the tme terval from the tme the system becomes empty utl the frst customer arrves ad eters the system. Defto 8. The server dle perod deotes the tme terval from the tme the system becomes empty utl the server begs to serve the frst customer. Let I j, l j,adb j deote the jth system dle perod, server dle perod, ad server busy perod tated wth the tal state (0 )( 0), respectvely; the, from the system assumptos, I j,j 1}, l j,j 1},adb j,j 1} are depedet radom varables, respectvely. Further, I k,k 1} follow a detcal geometrc dstrbuto wth mea 1/p. For l j,j 1}ad b j,j 1},thefollowgsobservedfrom assumpto (5). () If (0 )0,the where f(z) pz/(1 pz), z < 1, p1 p. () If (0 )>0,thel j I j τ 1 τ 1 ad b j b, j 1.Thus,thePGFsofl j ad b j j 1are f (z) ad b (z), z < 1,respectvely. Lemma 9 (see [19]). If z < 1, c >0,adc(z) 0 c z, the 1 lm (1 z) c (z) c< lm c z 1 k c<. (6) Lemma 10 (see [19]). If lm c c <, the lm (1/) c k c. 3. Aalyss of System Sze Dstrbuto Ithssecto,aboveall,wewlldscussthesystemsze dstrbuto at ay epoch server busy perod b.ext,the PGF of traset system sze dstrbuto at ay epochs s obtaed. Fally, the steady-state system sze dstrbuto s derved. l j I 1, j 1, I j τ 1 τ 1, j 2, b, j 1, b j b, j 2. (4) 3.1. Traset System Sze Dstrbuto Server Busy Perod b. Let Q j ( ) Pb >,( )j}, j 1, deote traset system sze dstrbuto at epoch server busy perod b ad q j (z) 0 Q j( )z, z < 1;theTheorem 11 follows.

4 4 Dscrete Dyamcs ature ad Socety Theorem 11. For j 1ad z < 1, q j (z) 1 G(pz) j 1 z C j 1 p j 1 p j1 [ G(pz pzb (z)) [ k1 g k z k r0 where p1 pad j 1 0f j 0. k1 j 1 q j g k (z) b 1 (z) 1 g k (pz pzb (z)) k C r k (pb (z))r p k r } ] }, ]} Proof. Deote by χ 1 thetotalgeeralzedservcetmeof the frst customer served server busy perod b, ad υ represets the arrvg umber of customers durg χ 1. otg that the legth of b has othg to do wth the servce order of customers ad arrval terval follows a geometrc dstrbuto wth parameter p,so we get P(V j) kj (7) g k C j k pj p k j, j 0. (8) Let A 1,A 2,...,A V be the customers who arrve the system durg χ 1, called class-1 customers. We call those who arrve after the class-1 customers as class-2 customers. Sce the legth of server busy perod s rrelevat to the servce order of customers, we adopt the followg servce order: class-1 customers are served the order of A 1,A 2,...,A V. After servg each class-1 customer, the server wll serve ay class-2 customer utl there are o class-2 customers preset. Thus the server busy perod b cabedecomposedas pot whe the server busy perod eds s a reewal pot, for j 1, Q j ( ) cabeexpressedas Q j ( )P χ 1 L 1 L V >,( )j} P χ 1 >,( )j} P χ 1 < χ 1 L 1 L V,( )j} P χ 1 >,j 1customers arrve (0, ]} k g k 0 C k p p k PL 1 L > ( k), (( k) )j}. (10) Ithelghtoftheaboveservceorder,ftheepoch( k) s L m (1 m )ad (( k) )j,theatepoch ( k) there are mclass-1 customers watg for servce ad j ( m) class-2 customers preset the system. By the law of total probablty ad otg that L m (1 m )has thesameprobabltypropertyasb,weget PL 1 L > ( k),(( k) )j} m1 m1 PL 1 L m 1 ( k) <L 1 L m, k lm 1 (( k) )j} PL 1 L m 1 l} PL m > ( k l), m1 (( k l) )j ( m)} k lm 1 PL 1 L m 1 l}q j m (( k l) ). Substtutg (11)to(10)leadsto Q j ( ) k1 g k C j 1 p j 1 p j1 (11) b χ 1 L 1 L V, (9) k g k 0 C k p p k (12) where L k,1 k V, deotes the tme terval from the epoch whe the server begs to serve the kth class-1 customer utl the ext epoch whe the servce of the (k1)th class-1 customer begs. Therefore, L k (k 1,2,...,V) are depedet radom varables wth the same dstrbuto as b. Obvously,L 1 L 2 L V 0whe V 0.Scethe m1 k lm 1 PL 1 L m 1 l} Q j m (( k l) ). Takg the PGF of (12) completes the proof of Theorem 11.

5 Dscrete Dyamcs ature ad Socety Traset System Sze Dstrbuto at Ay Epoch. Let P j ( ) P( )j (0 )},, j 0, be codtoal dstrbuto of traset system sze at ay epochs ad p j (z) 0 P j( )z, z < 1;theTheorem 12 follows. Theorem 12. For z < 1 ad 1, p 00 (z) 1 f(z) 1 z p 0 (z) f (z) b (z) 1 }, 1 [f(z) b (z)] [1 f (z)]b (z) (1 z) 1 [f (z) b (z)] }, (13) where f(z) pz/(1 pz) ad b(z) s determed by Lemma 6. Proof. otg that the system s empty at tme epoch f adolyfthesystemsthe systemdleperod, thuswe have Theorem 13. For z < 1, 1,adj1,2,..., 1, p 0j (z) f(z) q j (z) p j (z) ( 1 f(z) 1 z fj1 (z) b (z) f 1 (z) b (z) (1 [f(z) b (z)] ) 1, q j k (z) bk 1 (z) ( 1 f(z) 1 z fj (z) b (z) f (z) b (z) q j k (z) b k 1 (z)) q j k (z) b k 1 (z)) (16) (17) P 00 ( )P0 <I 1 } k2 k 1 P 1 PI 1 > } k2 u2k 2 (l b ) < k 1 P 1 PI k > ( u) }, k 1 1 (l b )u} (l b )I k } (14) where I, l,adb deote the th system dle perod, the th server dle perod, ad the th server busy perod, respectvely (see Deftos 5 8). For 1,smlarly,wehave (1 [f(z) b (z)] ) 1, where f(z) pz/(1 pz) ad b(z) ad q j (z) are determed by Lemma 6 ad Theorem 11,respectvely. Proof. For j 1,2,..., 1, ( )jmeas that epoch s server busy perod or server dle perod wth some customers watg for servce, so P 0j ( )PI 1 <I 1 b 1,( )j} j 1 PI 1 b 1 I 2 τ l <I 1 b 1 I 2 1 PI 1 b 1 I 2 j τ l } τ l,( )j} P 0 ( ) u Pb u}pi 1 > ( u) } k3 u2k 4 k 2 P 1 PI k 1 > ( u) }. (l b b )u} (15) u1 PI 1 u}q j (( u) ) uj2 j 1 PI 1 b 1 I 2 Pτ j > ( u) } u2 1 PI 1 b 1 I 2 τ l u} τ l u} Takg the PGFs of (14) ad(15), respectvely, ad usg the PGFs of I, l, b,adb,wecaobta(13). P j (( u) ). (18)

6 6 Dscrete Dyamcs ature ad Socety For 1,smlarly,weget P j ( ) uk 1 Pb k 1 u} Q j k (( u) ) where f(z) pz/(1 pz) ad b(z) ad q j (z) are determed by Lemma 6 ad Theorem 11,respectvely. Proof. For j,1,2,..., ( )jdcates that epoch s server busy perod, so we have uj j 1 Pb I 2 τ l u} (19) P 0j ( ) u1 PI 1 u}q j (( u) ) Pτ j > ( u) } u Pb 1 I 2 P j (( u) ). τ l u} Takg the PGFs of (18)ad(19), respectvely, gves rse to p 0j (z) f(z) q j (z) P j ( ) u2 1 PI 1 b 1 I 2 P j (( u) ), Q j k (( u) ) τ k } (24) p j (z) 1 f(z) 1 z fj1 (z) b (z) f 1 (z) b (z) p j (z), q j k (z) bk 1 (z) (20) 1 f(z) 1 z fj (z) b (z) f (z) b (z) p j (z), (21) whch leads to the followg relatoshp betwee p j (z) ad p 0j (z): uk 1 u Pb k 1 u} Pb 1 I 2 P j (( u) ), 1. τ k u} Takg PGFs of (24) ad performg smlar operatos the proof of Theorem 13, we ca complete the proof. p j (z) q j k (z) bk 1 (z) b 1 (z) f (z) p 0j (z) b 1 (z) q j (z), (22) 3.3. Steady-State System Sze Dstrbuto at Ay Epoch Theorem 15. Lettg p j lm P( ) j}, j 0, 1, 2,...,the 1,2,..., j1,2,..., 1. Substtutg (22) to(20) yelds(16), ad, furthermore, we obta (17). Theorem 14. For z < 1, 1,adj, 1,2,..., p 0j (z) f(z) q j (z) (1) for ρ pμ(1 αβ)/θ 1, p j 0, j 0, 1, 2,...; (2) for ρ pμ(1 αβ)/θ < 1, the steady-state system sze dstrbuto p j, j 0,1,2,...} s gve by the followg formulas: p j (z) f1 (z) b (z) q j k (z) bk 1 (z) 1 [f(z) b (z)], q j k (z) bk 1 (z) f (z) b (z) q j k (z) bk 1 (z) 1 [f(z) b (z)], (23) p 0 1 ρ, p j 1 ρ (1 p Δ j k ), p j 1 ρ p Δ j k, j1,2,..., 1, (25) j, 1,...,

7 Dscrete Dyamcs ature ad Socety 7 Δ j (3) where 1 G(p) C j 1 j 1 j 1 1 Δ 1 1 G(p) p G(p), Δ j [1 (4) whe j 0, j 1 0. p j 1 p j1 g k k1 g k k1 g k r0 C r k pr p k r ] } }, } j2,3,..., (26) Proof. Applyg 0 P j ( ) P(0 )}<1ad Lemmas 9 ad 10 yelds p j lm P j ( ) P (0 )} 0 0 P(0 )} lm P j ( )lm (1 z) p j (z).] z 1 (27) So p j,j 0}s obtaed by Lemma 6,Theorems11 14,ad L Hosptal s rule. Also, for ρ pμ(1 αβ)/θ < 1, j0 p j 1sclear oly by otg that j1 z j Δ j k z(1 z )[1 G(p pz)] p (1 z)[ G(ppz) z], Δ j k j1 ρ p(1 ρ). (28) Corollary 16. Let π (z) j0 p j zj, z < 1,deotethePGF of steady-state system sze dstrbuto p j,j 0}at ay epoch ;the,forρ pμ(1 αβ)/θ < 1, π (z) (1 ρ)(1 z) G(ppz) G(ppz) z 1 z (1 z). (29) Proof. Usg π (z) p 0 j1 p j zj ad (25), we obta (29). Remark 17. Equato (29) meas that the system sze L of - polcy Geo/G/1 queue wth feedback ad reparable server s the sum of two depedet radom varables: L L 0 L 1, f ρ pμ(1 αβ)/θ < 1. L 0 s the system sze of Geo/G/1 queuewthfeedbackadreparableserver,adl 1 s the addtoal system sze due to the -polcy. Ths cofrms that the stochastc decomposto property by Fuhrma ad Cooper [20] s also vald for the dscrete-tme queue uder cosderato. Corollary 18. Let E[L ] deote the expected steady-state system sze at ay epoch ;the,forρ pμ(1 αβ)/θ < 1, E[L ]ρ p2 G (1) 2(1 ρ) 1. (30) 2 Proof. Equato (30) s easly obtaed by E[L ] (dπ (z)/dz) z1. Remark 19 (specal case). If θ 1 ad α 0 (o feedback ad o server breakdow), the our model becomes a dscrete-tme -polcy Geo/G/1 queueg system. I ths case, for ρpμ<1,wehave π (z) (1 ρ)(1 z) G(ppz) 1 z G(ppz) z (1 z), E[L ]ρ p2 G (1) 2(1 ρ) 1, 2 (31) whch agree wth those [8](settgp1 (15), (16), ad (18) for [8] s model). 4. A umercal Example Our study has a potetal applcato a etwork access proxy system. I such a system, Chael requests, grats, data trasmssos, ad receptos all proceed fxed tme tervals. That s, servce request sedg, preprocessg, ad processg are doe a dscrete-tme maer. Servce requests set by the users ca be modelled as a Beroull process wth rate p. To avod frequet swtchovers from dle state to busy state, the proxy server s desged to start servg exhaustvely, whle the servce requests are accumulated to. If a servce request s receved wth errors at the destato, t s reset (feedback) wth probablty 1 θ. Whe the server s busy, the arrvg requests are placed queue. Theservcetmeofeachrequestsassumedtofollowa geometrc dstrbuto wth mea μ.also,the servce may be terrupted (server breakdow) due to some upredctable evets, such as traffc cogesto of the etwork. Suppose that upredctable evets occur accordg to a Beroull process wth rateα ad servce terrupto s mmedately recovered. The recovery tme obeys a geometrc dstrbuto wth mea β. The servce wll cotuously start whe the terrupto s recovered. umercal results Table 1 llustrate the applcato of the steady-state system sze dstrbuto system capacty desg. Here, we let p 0.15, μ2, θ 0.9, α 0.01, β5, ad 5.ByTheorem 15 ad Corollary 18,weumercally obta the steady-state servce request umber dstrbuto p j,j 0},themeaservcerequestumberE(L ),adthe system load ρ for the above etwork access proxy system (see Table 1).

8 8 Dscrete Dyamcs ature ad Socety Table 1: The steady-state servce request umber dstrbuto for a etwork access proxy system (p 0.15, μ2, θ 0.9, α 0.01, β5, ad 5). p 0 p 1 p 2 p 3 p 4 p 5 p 6 p p 8 p 9 p 10 p 11 p 12 p 13 p 14 p p 16 p 17 p 18 p 19 p 20 p 21 p 22 p p 24 p 25 p 26 p 27 p 28 p 29 p 30 p p 32 p 33 p 34 p 35 p 36 p 37 p 38 p E(L ) , ρ < 1, 39 j0 p j Wth Matlab 7.0, the data here are accurate for fve places of decmals. Table 2: The optmal system capacty of a etwork access proxy system for dfferet values of cotrollable parameter γ (p 0.15, μ2, θ 0.9, α 0.01, β5,ad5). γ M Deote by L the servce request umber the system. Accordg to Table 1,theprobabltesthatL exceeds E(L ) 1, E(L ),ade(l )1are obtaed as PL >E(L ) 1}1 PL E(L ) 1} 1 2 j0 p j , PL >E(L )} 1 P L E(L )} 1 3 j0 p j , PL >E(L )1}1 PL E(L )1} 1 4 j0 p j (32) The above three probablty values are greater tha 23%, whch shows that t s qute accurate for system capacty desg to use mea servce request umber E(L ) because qute a few servce requests wll be lost due to o watg space. Also, t s see from Table 1 that the probablty that the servce request umber the system exceeds 29 s less tha 0.01%. So t s also uecessary for ths system to desg alargecapacty. Let M be system capacty ad let the postve decmal γ be less tha 1. For smplcty, we assume that system capacty M takes postve teger value. Whe the overflow probablty P(L > M) γ,fromtable 1,wecagetoptmalsystem capacty M so that system operatg costs are mmzed (see Table 2). For example, f γ , thetfollows from P(L > M) γ ad Table 1 that M 31.Thuswe take M 31sce small capacty leads to small system cost. 5. Coclusos I ths paper, we cosder a dscrete-tme -polcy Geo/G/1 queueg system wth feedback ad reparable server. Wth a probablty aalyss ad reewal process theory, we dscuss the traset system sze dstrbuto ad derve the steadystate system sze dstrbuto ad ts PGF. It should be oted that the steady-state system sze dstrbuto derved by ths paper s qute sutable for umercal calculato. umercal examples show the applcato of steady-state system sze dstrbuto system capacty desg for a etwork access proxy system. I the future, further study, such as the radom -polcy Geo X /G/1 queueg system wth feedback ad reparable server, wll be the research topc wth the help of smlar deas ad methods. Coflct of Iterests The authors declare that there s o coflct of terests regardg the publcato of ths paper. Ackowledgmets The authors would lke to thak the referees ad the edtor for ther valuable commets ad suggestos. Ths work s supported by the Basc ad Froter Research Foudato of Chogqg of Cha (cstc2013jcyja00008) ad the Scetfc Research Startg Foudato for Doctors of Chogqg Uversty of Techology (2012ZD48). Refereces [1] H. Kobayash ad A. G. Kohem, Queueg models for computer commucatos system aalyss, IEEE Trasactos o Commucatos,vol.25,o.1,pp.2 29,1977.

9 Dscrete Dyamcs ature ad Socety 9 [2] I. Rub ad L. F. M. De Moraes, Message delay aalyss for pollg ad toke multple-access schemes for local commucato etworks, IEEEJouraloSelectedAreas Commucatos,vol.1,o.5,pp ,1983. [3] F. J. L ad R. H. Su, Measuremet-estmato approach to effcecy evaluato of badwdth allocato scheme ATM etworks, Proceedgs of the Coferece Record IEEE Iterato Coferece o Comucatos,vol.6,o.2,pp , [4]. S. Ta, X. L. Xu, ad Z. Y. Ma, Dscrete-Tme Queueg Theory, Scece Press, Bejg, Cha, [5] M. Yad ad P. aor, Queueg systems wth a removable servce stato, Operatoal Research Quarterly, vol.14,o.3, pp ,1963. [6] L. Tadj ad G. Choudhury, Optmal desg ad cotrol of queues, Top, vol. 13,o. 2, pp , [7] P. Moreo, A dscrete-tme sgle-server queue wth a modfed -polcy, Iteratoal Joural of Systems Scece, vol.38,o. 6, pp , [8] T.-Y. Wag ad J.-C. Ke, The radomzed threshold for the dscrete-tme Geo/G/1 queue, Appled Mathematcal Modellg, vol.33,o.7,pp ,2009. [9] Y. Y. We, M. M. Yu, Y. H. Tag, ad J. X. Gu, Queue sze dstrbuto ad capacty optmum desg for -polcy Geo (λ 1,λ 2,λ 3 ) /G/1 queue wth setup tme ad varable put rate, Mathematcal ad Computer Modellg, vol.57,o.5-6,pp , [10] T.-Y. Wag, Radom -polcy Geo/G/1 queue wth startup ad closedow tmes, Joural of Appled Mathematcs,vol.2012, Artcle ID , 19 pages, [11] L. Takács, A sgle-server queue wth feedback, The Bell System Techcal Joural,vol.42,o.2,pp ,1963. [12] B.D.Cho,B.Km,adS.H.Cho, OtheM/G/1 Beroull feedback queue wth mult-class customers, Computers & Operatos Research,vol.27,o.3,pp ,2000. [13] D. I. Cho ad T.-S. Km, Aalyss of a two-phase queueg system wth vacatos ad Beroull feedback, Stochastc Aalyss ad Applcatos,vol.21,o.5,pp ,2003. [14] B. Krsha Kumar ad J. Raja, O multserver feedback retralqueueswthbalkgadcotrolretralrate, Aals of Operatos Research,vol.141,pp ,2006. [15] I. Ateca ad P. Moreo, Dscrete-tme Geo [X] /G H /1 retral queue wth Beroull feedback, Computers & Mathematcs wth Applcatos,vol.47,o.8-9,pp ,2004. [16] J. Wag ad P. Zhag, A dscrete-tme retral queue wth egatvecustomersadurelableserver, Computers & Idustral Egeerg,vol.56,o.4,pp ,2009. [17] Y.H.Tag,X.Yu,adS.J.Huag, Dscrete-tmeGeo X /G/1 queue wth urelable server ad multple adaptve delayed vacatos, Joural of Computatoal ad Appled Mathematcs, vol. 220, o. 1-2, pp , [18] Z. M. Lu ad S. Gao, Relablty dces of a Geo/G/1/1 Erlag loss system wth actve breakdows uder Beroull schedule, Iteratoal Joural of Maagemet Scece ad Egeerg Maagemet,vol.5,o.6,pp ,2010. [19] J. J. Huter, Mathematcal Techques of Appled Probablty, Dscrete Tme Models: Techques ad Applcatos, vol. 2 of Operatos Research ad Idustral Egeerg, Academc Press, ew York, Y, USA, [20] S. W. Fuhrma ad R. B. Cooper, Stochastc decompostos the M/G/1 queue wth geeralzed vacatos, Operatos Research, vol. 33, o. 5, pp , 1985.

10 Advaces Operatos Research Advaces Decso Sceces Joural of Appled Mathematcs Algebra Joural of Probablty ad Statstcs The Scetfc World Joural Iteratoal Joural of Dfferetal Equatos Submt your mauscrpts at Iteratoal Joural of Advaces Combatorcs Mathematcal Physcs Joural of Complex Aalyss Iteratoal Joural of Mathematcs ad Mathematcal Sceces Mathematcal Problems Egeerg Joural of Mathematcs Dscrete Mathematcs Joural of Dscrete Dyamcs ature ad Socety Joural of Fucto Spaces Abstract ad Appled Aalyss Iteratoal Joural of Joural of Stochastc Aalyss Optmzato