Queueing Anlysis of Erly Messge Discrd olicy rijt Dube Eitn Altmn ; Abstrct We consider in this pper pckets which rrive ccording to oisson process into finite queue. A group of consecutive pckets forms frme (or messge) nd one then considers not only the qulity of service of single pcket but lso tht of the whole messge. In order to improve required qulity of service, either on the frme loss probbilities or on the dely, discrding mechnisms hve to be used. We nlyze in this pper the performnce of the Erly Messge Discrd (EMD) policy t the buffer, which consists of () rejecting n entire messge if upon the rrivl of the first pcket of the messge, the buffer occupncy exceeds threshold, nd () if pcket is lost, then ll subsequent rrivls tht belong to the sme messge re discrded. Index Terms EMD policy, pcket model, queue-length distribution, goodput. qλ λ λ λ pλ pλ pλ pλ o o o 0,0,0,0 3,0 4,0 5,0 Ν,0 µ µ µ µ µ µ µ qλ qλ qλ qλ qλ 0,,, 3, 4, 5, o o o µ µ µ µ µ µ µ N, λ I. INTRODUCTION Quite often qulity of service hve to be studied with respect to not only single pcket, but to whole messge or frme. For exmple, in ATM trnsport lyer protocol (AAL) is responsible for grouping pckets into frme, nd lost pcket implies the corruption of the whole frme. Selective Messge Discrding (nd EMD in prticulr, on which we focus here) hve been proposed to chieve the twin gols of incresed goodput nd reduced network congestion by discrding the pckets which do not belong to (or hve potentils of not belonging to) good messges ( messge is good if it is entirely received t the destintion). Rejecting entire messges could lso serve to gurntee n cceptble verge dely bound for ccepted messges. The gol of this pper is to present explicit expressions for the queue-length distribution nd the goodput (defined s in [0] s the rtio between totl pckets comprising good messges exiting the network node nd the totl rriving pckets t the input). Our strting point is the Mrkovin model proposed in [0]: oisson process of pcket rrivls, geometriclly distributed frme size, nd exponentilly distributed service times of pckets. In [0], recursive procedures hve been proposed for the computtion of the performnce mesures, but explicit expressions hve not been obtined. Our nlyticl results on closed form expressions for performnce metrics (in prticulr the queue-length distribution nd the goodput) my be quite useful in dimensioning the buffer size tht should be used for given goodput, in the study of the sensitivity of the goodput to different prmeters for e.g., the messge length, the buffer size, the lod nd most importntly in finding n estimte of the optiml discrding threshold etc. In previous work [5], we nlyzed the rtil Messge Discrd (MD) policy in which only if some pcket of messge is lost, subsequent pckets re rejected (but entire messges re not discrded, in contrst with EMD). As the pcket level nlysis turns to be quite complex nd involved, we studied in [3], [4], [5], [6] some fluid pproximtions. Some other references on numericl studies of MD nd EMD policies re [8], [7]. INRIA, B.. 93, 0690, Sophi Antipolis Cedex, Frnce. E-mil: frijt.dube,eitn.altmng@sophi.inri.fr. CESIMO, Universidd de Los Andes, Fcultd de Ingeneri, Merid, Venezuel Fig.. Trnsition structure under the EMD policy In Section II we describe our queueing model nd present our min results on the z-trnsform of the queue-length distribution nd then the explicit expressions for the stedy-stte probbilities. In Section III we present n pproch for obtining the explicit expression for the goodput rtio using lgebric techniques. II. ACET MODEL The pcket model is the sme s the one proposed in [0]. We first describe the model in brief. In terms of pcket the network element is M=M==N queue with rrivl rte nd service rte μ nd the lod = μ. A messge length (in terms of pckets) is considered to be geometriclly distributed with prmeter q. Under the EMD policy, threshold level ( is n integer, 0 N) is fixed. If messge strts to rrive when the buffer occupncy is t or bove pckets, then ll the pckets of tht messge re discrded. Also, if pcket belonging to n ccepted messge is discrded due to buffer overflow then ll the subsequent pckets belonging to the sme messge re lso discrded. To model the policy, two modes for working of the network element re defined: the norml mode, in which pckets re dmitted, nd the discrding mode, in which rriving pckets re discrded. The stte trnsition digrm for EMD policy under this model is shown in Figure (). Let i;j(0 i N;j =0; ) be the stedy-stte probbility of hving i pckets in the system nd the system is in mode j (j =0fornorml; j =for discrding). We now define the trnsform functions A j(z) = zi N i;j, B j(z) = i= zi i;j nd Q j(z) =A j(z)b j(z) for j =0;. A. GF nd distribution of the number of pckets in the queue roposition : The probbility generting functions A j(z) nd B j(z) for j =0; re given by, A 0(z) = (q z )( 0;0( z ) ;0z ;0z ) ;qz 0-7803-7400-/0/$7.00 (C) 00 IEEE
where, 0;( z )qz Λ D A (z) = 0;( z ) ;(q )z (q z ) B 0(z) = ;0rz N;0rz N ;0z ( rz z ) B (z) = ;(z rz ) ;0q N;0z N (( z )( q) rz) ;0q( z )Λ z D D = ( z z )(q z ) D = ( rz z )( z ) ;0 = 0; = q( ) D q( q) 0;0 = ;0 ; = rn N ;0 rn N ;0 rn N q( q) N;0 = (WW)N (W W )r ;0 N ;0 = rn ;0 N D = q ( ) rn N q( q) rn N ( )( q) (q ) ( )( q) (q ) q( q) ( )( q)r 3 N (W W ) N (W W ) ( q) ( q ) rn N ( q q) ;0 r rn N with W ; = ()±(() 4r) r nd y = W y W y for ny y. roof: Refer to Appendix A. Corollry : The trnsition probbilities re given by, For i, (i) i;0 = ;0 i i (i) ;q ( q) (i) ( )( ( q) ) N;0 Ni ( q) (i) ; ( ( q) )( ( q) ) i; = ;(( q) i ) N;0 ( Ni Ni) ;0 ;0 And for i N, i ri i ri i;0 = N;0Ni ; i; = N;0(NiNi) where =. roof: Refer to Appendix B. Hving obtined the explicit expressions for the sttionry distribution of the queue-length we next proceed to obtin the expression for the goodput rtio (s defined in Section I). III. GOODUT RATIO Let W be the rndom vrible tht represents the length (number of pckets) of n rriving messge. Let V be the rndom vrible representing the success of messge, V = for good messge, nd V = 0 for messge which hs one or more dropped pckets. Then G cn be expressed s (see [0]) q n= nq( q)n N : (V =jw = n; Q = i) (Q = i) 4 Denote the conditionl probbilities S n;i = (V =jw = n; Q = i). In [0], recursions for evluting these probbilities nd hence G were given. We will present here n explicit expression for G. Todo this we will use the multidimensionl generting function for probbilities S n;i which ws obtined in different context in [] nd in [9]. We define the two-dimensionl generting function of S n;i for n nd 0 i N, ss(x; μ y), i.e., S(x; μ y) = N n= Sn;ixn y i, We will next reproduce the roposition we developed in [5] for the cse of rtil Messge Discrd(MD) policy. A MD is n EMD with >N. roposition : The probbility generting function μ S(x; y) cn be expressed s S(x; μ N y) = ci(x)yi where for 0 i N, c i(x) is equl to nd c N (x) =0with, 3 A A y N(i) A 3y N(i) 4 By Ni B y Ni y ; = ± p ( ) 4x 3 = x 4 = x(y N y N ) y N (y ) y N (y ) A = =(( y )( y )) A = =(( y )(y y )) A 3 = =(( y )(y y )) B = B = (y y ) 0-7803-7400-/0/$7.00 (C) 00 IEEE
For the EMD policy, the conditionl probbilities S n;i re sme s tht for the MD policy for i<. If the hed of the messge rrives when the system occupncy is t or bove the threshold the messge s whole is rejected. Thus for the EMD policy we define the trnsition probbility generting function s S(x; μ y) = ci(x)yi,where c i(x) is given by roposition (). And the expression for Goodput rtio cn be expressed (like in roposition (3) in [5], with N in the summtion replced by ) s, G = q d(xci(x)) (Q = i) dx x=(q) ψ ψ!! = q 4 d ( q) c i(x) (Q = i) dx c i( q) (Q = i) # x=(q) where sttionry probbilities (Q = i) = i;0 i; re known from Corollry. A. Exct expression for G In this section we im to derive n exct expression for G for EMD policy. From () we find tht we need n expression for ci(x) (Q = i). Observe tht ci(x) (Q = i) = c 0(x) (Q =0) ci(x) (Q = i). Wehve (Q =0)= i= 0;0 0; with 0;0 nd 0; given by Corollry. We next find generl expression for (Q = i) for i =to. From Corollry, by dding i;0 nd i; for i =to we cn express (Q = i) s, (i) ;0 r i Ni( ) Ni N;0 ; (i) q ( )( q ) ( q)(i) ( q ) ( q) i ;0 i r r i We will now find n expression for ci (Q i= = i).from() nd roposition we write fter some lgebr (see [] for detils), i= ci (Q = i) s ;0 ; ;0 c i ;0 i= We shll not explicitly show x in prentheses for function c i i= () () c i i i= i= c i i ;q ( )( q ) ;0() c i( q) i ;( q)( q) q ;0ry N;0y N (y ( ) ) yr r i= c iy i i= c iy i ;0y ;0ry N;0yN (y ( ) ) (3) yr ;0y r Observe tht the lst expression contins terms of the form i= cii (with = ;y ;y ;;( q)). We now obtin these terms from the expression for c i from roposition. For ny, i= cii is equl to ( )( ( 3A ) ) ( 4B y N 3A y N y y ) y ( 4B y N 3A 3y N y y ) y Thus fter some rerrngements we cn express the expression for 3 ci (Q = i) from (3) s i= ( )( 3A )[ 5 6 9 F F ]( 4B y N 3A y N )[F 3 F F 4 F 5 F 6]( 4B y N 3A 3y N ) [F 7 F F 8 F 9 F 0] where, 5 = ( ) 7; 6 = (q)((q) ) (q) 8 F = E (y )E (y ); F = E (y )E (y ) F 3 = E (y =) 7; F 4 = E (y =( q)) 8 F 5 = E (y)e (y ); F 6 = E (y y )E (y ) F 7 = E (y =) 7; F 8 = E (y =( q)) 8 F 9 = E (y y )E (y ); F 0 = E (y)e (y ) F = E (y ); F = E (y ) with, E (y) = y (y ) ) nd E y (y) is equl to ;0ry N;0yN (y( ) ) yr ;0y r 3 It should be noted tht A ;A ;A 3 ;B ;B ; 3 nd F i ;i=; ;:::;: re ll functions of x. 0-7803-7400-/0/$7.00 (C) 00 IEEE
nd, 7 = 8 = ; ( )( q ) ;0() ;( q)( q) q 9 = ( ) ;0 ( ) ;q 0;( )q =0 (4) ;0rW N;0rW N ;0W =0 (5) ;0rW N;0rW N ;0W =0 (6) with W ; s defined in roposition. We lso hve N (i;0 i;) =which is sme s A 0() B 0() A () B () = (7) Hving obtined n expression for ci (Q = i), one cn directly obtin the expression for G from () (however, we lso need i= derivtive of ci (Q = i) with respect to x which is esy to i= obtin). IV. CONCLUSION We provided explicit expressions for the sttionry distribution of the queue-length nd the goodput for the EMD policy. An interesting extension will be to study the symptotic behvior of EMD policy, either from the generting functions or from the explicit expressions nd to obtin simpler pproximtions vlid for symptotic regimes (lrge buffer, hevy trc etc). AENDI A We hve the following set of equtions from [0] (with r = q). 0;0 = ;0 (4) q 0; = ; (5) ( ) i;0 = i;0 i;0 q i; for i (6) ( ) i;0 = r i;0 i;0 for i N (7) ( ) N;0r = N;0 (8) N; = N;0 (9) (q ) i; = i; i (0) i; = i; q i;0 i N () Tking z-trnsforms of (6), (7), (8) nd (0) nd from (4) (5) nd (9) fter some lgebr we get A j(z) nd B j(z) s in roposition in terms of 0;0; 0;; ;0; ;0; ; nd N;0. Next observe tht ll the trnsform functions A j(z);b j(z), j = 0;, re polynomil in z (N is finite) nd hence nlytic. Thus the numertor of the right hnd side of the expressions for A j(z);b j(z) vnishes t the zeros of the denomintor of the right hnd side. Thus, substituting the zeros of the denomintor in the numertor nd equting it to 0 in the expressions for A j(z);b j(z) we get the following set of five independent equtions: q( ;0 ;0) ;q = 0 ; ;0 = ;0 () 0;q( q) = ; (3) (q ) 0;0( ) ;0 ;0 Since, A 0(z) nd B (z) hve the form of 0=0 t z =we shll tke, A 0() = lim z! A 0(z) nd B () = lim z! B (z). Thus, we hve the following eqution from (7) q( ) = q( 0;0 0;) ( q q) ;0 ( q ) ( )( q) N;0 ;0 ( q) ; (8) Thus, the six unknowns 0;0; 0;; ;0; ;0; ; nd N;0 cn be obtined by solving the six equtions, (), (3), (4), (5), (6) nd (8) in six unknowns. We hve, Q 0(z) = A 0(z) B 0(z) = AENDI B 0;0( z ) ;0z ;0z ( z z ) ;qz 0;( z )qz ( z z )(q z ) ;0rz N;0rz N ;0z ( rz z ) Grouping the terms with the sme constnts of the type i;j we express the expression for Q 0(z) in the following formt, 0;0 Q 0(z) = (z ) ;0z (z )(z ) ;0z (z W )(z W ) r(z W )(z W ) (z )(z ) N;0z N ;z (z W )(z W ) z ( q)(z )(z ( q)) (z ) 0; (9) Applying prtil frction pproch to the lst eqution nd fter some rerrngements (see [] for detils) we get Q 0(z) equl to, z ;0 z () ( ) z z z W z W (W W ) z W z W 0-7803-7400-/0/$7.00 (C) 00 IEEE
;0 z W z W r(w W ) z W z W r(w W) z z () ( ) z z N;0 z N W N zn W N (W W ) z W z W ;q ( q) z ( )( ( q) )(z ) (z () ) ( )( ( q) )(z ) z ( q) () ( ( q) )( ( q) )(z ( q) ) 0;q z ( q)( ( q) ) z z ( q) z ( q) We cn now find the inverse trnsform of Q 0(z) nd obtin the stedystte probbilities i;0 for 0 i. This cn be esily done s the lst equlity contins terms of the form zm m. Thus, we by the z inverse z-trnsform of lst eqution, we get the expression for i;0 s in Corollry. The expression for i; is obtined by finding the inverse z-trnsform of Q (z) long similr lines. We re not providing here the detils which cn be found in []. REFERENCES [] Eitn Altmn nd Aln Jen-Mrie. Loss robbilities for Messges with Redundnt ckets Feeding Finite Buffer. IEEE Journl on Selected Ares in Communictions, 6(5):764777, June 998. [] rijt Dube nd Eitn Altmn. Queueing nd Fluid Anlysis of Erly Messge Discrd olicy. extended version vilble t http://wwwsop.inri.fr/mistrl/personnel/rijt.dube/. [3] rijt Dube nd Eitn Altmn. On Fluid Anlysis of Queues with Selective Messge Discrding olicies. In proc. of the 38th Annul Allerton Conf on Comm., Control nd Computing, Illinois, USA, October 000. [4] rijt Dube nd Eitn Altmn. On the Worklod rocess in Fluid Queue with Bursty Input nd Selective Discrding. In proc. of the 7th Interntionl Teletrc Congress, Slvdor d Bhi, Brzil, September 4-8 00. [5] rijt Dube nd Eitn Altmn. Queueing nd Fluid Anlysis of rtil Messge Discrd olicy. In proc. of 9th IFI Working Conference on erformnce Modelling nd Evlution of ATM nd I Networks, Budpest, Hungry, June, 7-9 00. [6] rijt Dube nd Eitn Altmn. Fluid Anlysis of Erly Messge Discrding olicy Under Hevy Trc. In proc. of IEEE INFOCOM 00, New York City, USA, June 00. [7]. whr,. itjim, T. Tkine nd Y. Oie. cket Loss erformnce of Selective Cell Discrd Schemes in ATM Switches. IEEE Journl of Selected Ares in Communictions, 5(5):90393, June 997. [8] Y. H. im nd S. Q. Li. erformnce Anlysis of Dt cket Discrding in ATM Networks. IEEE/ACM Trnsctions on Networking, 7():6 7, 999. [9] Omr Ait-Helll Eitn Altmn Aln Jen-Mrie Irin A. urkov. On loss probbilities in presence of redundnt pckets nd severl trc sources. erformnce Evlution, 36-37:48558, 999. [0] Yel Lpid Rphel Rom nd Moshe Sidi. Anlysis of Discrding olicies in High-Speed Networks. IEEE Journl on Selected Ares in Communictions, 6(5):764777, June 998. 0-7803-7400-/0/$7.00 (C) 00 IEEE