Semi-analytic Evaluation of Quality of Service Parameters in Multihop Networks

Transcription

1 U J.T. (4): -4 (pr. 8) Sem-analyt Evaluaton of Qualty of Serve arameters n Multhop etworks Dobr tanassov Batovsk Faulty of Sene and Tehnology, ssumpton Unversty, Bangkok, Thaland <dbatovsk@au.edu> bstrat The performane evaluaton of large multhop networks n terms of the standard qualty of serve parameters suh as lateny, throughput, network utlzaton, et., s often performed wth the use of dsrete-event smulaton tools whh would requre substantal omputatonal resoures wth the nrease of the number of nodes n the network. n alternatve approah s onsdered n ths ontrbuton whh s based on the analytal estmaton of the loal queung performane for a gven set of nput parameters ombned wth omputatonal experments for statstal evaluaton of the ontrol varables for the entre network. Ths sem-analytal approah proves useful for arbtrary teleommunaton networks where varous traff and serve ondtons an be tested under dfferent routng senaros wth mnmal developmental efforts. Sample open network onfguratons are onsdered as an llustraton of the statstal evaluaton of the network performane. Keywords: Multhop networks, general queung, performane evaluaton.. Introduton The multhop networks onsst of nteronneted nodes whh relay dgtal nformaton enapsulated n pakets of varable length. The smultaneous arrval of bursts of pakets at nodes wth lmted transmsson apabltes may trgger a ongeston resultng n nreased lateny and redued throughput. The management of paket flows depends on a number of bas fators, suh as traff pattern, rate onstrants, network topology, node moblty, node desgn, et. The node desgn s of a partular nterest as t restrts the analytal onsderatons wthn the hardware lmtatons of real systems, for example, wreless nodes. The evoluton of network deves s based on the balane between smplty, safety and effeny. Wreless deves ommonly utlze a sngle hannel for ommunaton wth a entralzed network. In order to funton n a prospetve deentralzed network, however, the sad sngle hannel may operate as a pat of a multple-nput sngleoutput (MISO) system. resumably, a wreless node reeves pakets from several adjaent nodes and then uses ts sngle hannel to further relay the paket flows to other onneted nodes. The store-and-forward prnple of operaton requres a buffer beng able to store the pakets durng bursty perods. The sngle hannel s atvated asynhronously for only one output port at a tme and the mean transmsson rate of the port depends on the sgnal-to-(nose-plus-nterferene) rato. t the nput, the number of smultaneously arrvng paket bursts an be redued wth a proper seleton of the atve nput ports at eah node. The resultant network onfguratons an be tested wth omputer smulatons, but ths s a tme onsumng approah. Fast omputatonal experments for a rapd evaluaton of network senaros for dfferent topologes and traff patterns an be performed nstead. Ths ontrbuton s wrtten as an attempt to ombne omputatonal tehnques wth exstng analyt results for the statstal analyss of multhop networks. Regular aper

2 U J.T. (4): -4 (pr. 8). Traff attern. nalyt Model Consder a set of ooperatve nodes beng nteronneted n an arbtrary multhop network. Eah node sends/reeves pakets of varable length to/from other nodes. Under the assumpton of a steady state, a statstal representaton of the paket exhange s gven by a traff pattern desrbed by the mean soure rates, [pakets/se], and the dmensonless squared oeffents of varaton (sv),,, from soure node s to destnaton node d; s, d,,,. The ndex p s used for a gven path, p,,,, where s the number of hosen paths for eah souredestnaton par s d. By defnton,, σ, () T,,. p where σ, and T, /, are the standard devaton and the mean value of the dstrbuton of nter-arrval tmes of onseutve pakets, orrespondngly. paket flow for a gven path of a soure-destnaton par s represented by a separate paket lass. The total number of dstnt lasses,, s obtaned by summaton, s d s d C C. () paket flow (lass) s haraterzed by ts own paket length dstrbuton. The sad dstrbuton s further onvoluted wth the serve dstrbuton (defned for a fxed mnmum paket length) of the output port at eah relayng lnk j to obtan an effetve serve dstrbuton desrbed by ts mean value, μ, and sv, μ,. traff pattern s defned as a set of mean and sv values over hosen paths for dfferent transmsson senaros: many-tomany, gateway, or unast paket flows. lso, broadast and musltast senaros an be onstruted wth a replaton of the paket flow at eah splttng pont, where a repla has the same mean rate as the ntal flow and the sv s obtaned from the flow statsts. mxed traff pattern may nlude a ombnaton of the aforesad routng senaros.. ath Seleton The soure-destnaton paths are usually determned by a hop ount so that the number of hops s mnmzed. The shortest paths depend on the network topology and a souredestnaton par may have a set of alternatve shortest paths for the same number of routng steps (hops). However, non-shortest paths may also prove useful n a ongested network. In realty, a hosen path s preferably the one wth the shortest round-trp delay at the end of the node dsovery. However, the round-trp delay depends not only on the number of hops but also on the ongeston level due to ross-traff and the varyng transmsson apabltes at ntermedate nodes. Durng omputatons, the path unertanty problem s resolved teratvely. The omputatonal experments are ntated wth randomly hosen shortest paths and a further refnement of the optmal paths takes plae. The omputaton tme depends on the qukaton of the algorthm for obtanng the prospetve soure-destnaton paths. n exhaustve searh proedure wth loop avodane s the default hoe for trang multple paths on the bass of a onnetvty matrx whh desrbes the onnetons between adjaent nodes. Startng from the soure node, the proedure expands the searh among frst neghbors, seond neghbors,, et., untl the destnaton s found. The effetveness of the algorthm depends on the mean node degree and the network dameter. Hopefully, most real networks are nonhomogeneous and the number of alternatve paths s qute lmted due to redued node degrees and topologal bottleneks. Ths smplfes and speeds up the ombnatoral problem n fndng a set of paths for eah soure-destnaton par. ote that nonoverlappng dstnt paths are the preferred hoe for multpath transmsson wth load balanng but most atual paths partally overlap eah other. Regular aper 6

3 U J.T. (4): -4 (pr. 8).3 ode Desgn The hoe of all-port or sngle-port mode of operaton depends on the avalable hannels per node. ssumng hgh sgnal-tonose ratos durng transmsson, wred multhop networks usually mplement the allport onfguraton where a node smultaneously sends/reeves pakets to/from many adjaent nodes beng multplexed wth the use of a swthng matrx n an nternal nteronneton network. But n wreless networks the low sgnal-to-nose ratos redue the transmsson rates and the smplfed allport mplementaton for wreless nodes may have the followng bas omponents: - Frst-ome frst serve (FCFS) queue for nomng pakets and a fast server at the network layer, Open Systems Interonneton (OSI) layer 3 (ISO ). - FCFS queues (storng pakets omng from the server at the network layer) and dstnt servers for eah output port at the data lnk layer (OSI layer ). The output ports are atvated smultaneously and the queue for eah output port s served ndependently from the other output ports. It s assumed that the routng desons made by the server at the network layer are performed muh faster than the transmssons of pakets by the output ports and the average queue length at the network layer s rather small ompared to the average queue lengths at the data lnk layer. In a more realst sngle-port senaro, wreless nodes reeve pakets from adjaent nodes but have only one transmsson hannel due to lmtatons of hannel reuse and power onsumpton. The sngle-port mplementaton has the followng bas omponents: FCFS queue for the nomng pakets from all nput ports and a sngle hybrd server performng the ombned onseutve funtons of both the network layer and the data lnk layer. The sngle hannel s used asynhronously by all output ports but whenever a partular output port s atvated, the remanng ports are n stand-by mode. The transmsson rates for dfferent output ports depend on the hannel ondtons to the adjaent nodes. Therefore, an output port s desrbed by ts own serve-tme dstrbuton. For both all-port and sngle-port ases, one an assume suffent buffer lengths whh would prevent automat paket drops as a result of buffer overflow. aket drop s expeted to our oasonally at a server on the bass of the statstal nformaton of paket arrvals aumulated for a suffent perod of tme ountng several tme ntervals wth empty queue. The mean queue length and the mean watng tme n a queue depend on the server utlzaton, ρ /μ. It s ommonly stated that the ongeston bulds up f ρ >.4. Dependng on a rtal value,.4 ρ MX.9, a paket drop s ntated at the server. s smaller ρ MX s, as smaller the mean delay would be, but the throughput would derease as well. On the ontrary, the nrease of ρ MX would mprove the throughput but substantally nrease the delay. Thus, ρ MX s an nput parameter whh sgnfantly affets the overall network performane. If the real-tme (RT) traff domnates, ρ MX must derease to satsfy the end-to-end lateny requrements. The non-realtme (RT) traff tolerates the nrease of ρ MX whh would form long queues durng bursty perods. s an alternatve to the undesrable paket drop durng ntermedate routng steps, the proper end-to-end ongeston ontrol at the transport layer (OSI layer 4) effetvely redues the soure rates. The ongeston ontrol algorthms are desgned to mantan optmal throughput and derease the end-toend delay. For UD pakets or TC pakets wthout ongeston ontrol, the rates are redued and the pakets are dropped at the ongested node. Congeston ontrol appled to TC pakets redues the rates at the soure nodes and the redued rates derease the server utlzaton and the lateny at the ntermedate relayng nodes between the soure node and the ongested node. Durng omputatonal experments, eah traff flow s assgned a ongeston ontrol parameter, C COGESTIO, ndatng the porton of the soure rate that an be redued at the soure to avod paket drops at a ongested node (hot spot). Regular aper 7

4 U J.T. (4): -4 (pr. 8).4 Iteratve roess The remanng text deals wth sngle-port node desgn. The nter-arrval tme dstrbutons from a node to the next node j are desrbed by unknown mean rates, [pakets/se], and squared oeffents of varaton,,,, j,,,. The serve-tme dstrbutons are desrbed by known mean rates, μ [pakets/se], and squared oeffents of varaton,,,, j,,,. The values of and, are obtaned n two onseutve steps: a) The rates are omputed teratvely on the bass of the overlappng and onsequent reduton of the ntal soure rates over the hosen paths. b) The squared oeffents of varaton are omputed teratvely usng the standard method for deomposton of open non-produt form networks n three onseutve steps: mergng, flow and splttng (Bolh et al. 998). In ths ontrbuton, the sad method s modfed to assgn a lass to the paket flow of eah path. The rate reduton at eah ntermedate node,, depends on spefed server utlzaton thresholds,.4 ρ MX,. 9, so that the node utlzaton, p, ρ (3) μ s, d, p, j should be lmted to ρ ρ MX,, (4) assumng a sngle server per node. n teratve proess s appled to determne optmal soure rates,, for μ whh no paket drop ould our at ntermedate nodes. The number of teratons depends on the maxmum number of routng steps for the longest path n the network. Durng the frst teraton, a reduton of ntal soure rates,,, s performed after overlappng the rates for all paths wthn the frst routng step. Durng the seond teraton, a further reduton s performed after overlappng the rates for all paths for two routng steps. The rate reduton ontnues onseutvely for an nreased number of teratons untl no further orreton s needed. The rates derease monotonally after every teraton. It should be mentoned that dfferent strateges for rate reduton ould be mplemented f needed. One the mean rates,, are known, the throughput for every path, p, p,,, s gven by: Th, p, () The throughput per soure-destnaton par s also obtaned: Th p. (6) soure-destnaton par an have several paths of varable lengths. However, t s useful to know the average throughput n the network for all paths,, h, havng a h presrbed number of hops, h h, as gven by: Th, h, p h, h h s d p s d h, p h. (7) The average throughput n the network for all paths,, s gven by Th s d s p d. (8) The seond teratve proess, for the omputaton of, for a number of output ports, J j, may use the followng formulas modfed from Bolh et al. (998), see also ujolle and Wu (986), Whtt (983, 983b): - Mergng from a for a number of nput ports, J k :, - Flow: J k s d s p d k (9), k k Regular aper 8

5 U J.T. (4): -4 (pr. 8) + ρ ( ) + ( ρ )( ) () D, S,, - Splttng to a number of output ports, J j :, + ( D, ) () wth: J j μ ( S, μ, ) s d p j μ s d () and μ J, (3) j μ s d p j s d, where: s the sv of nter-arrval tmes; s the sv of nter-departure tmes; and S, D, the sv of serve tmes (see also Eqs. (.39) and (.4) n Bolh et al. 998). The teratve proess for the omputaton of the unknown oeffents,, starts, from the default ntalzaton,,, and ontnues untl no further mprovement n the onvergene proess s observed for a requred number of true dgts. The modfed mergng-flow-splttng tasks desrbed by Eqs. (9-) are performed for every dstnt soure-destnaton path (ndexes s, d, and p) and every output port (ndexes and j) on the way from soure to destnaton. The omputatonal omplexty arses from two ssues whh have to be addressed durng mergng and splttng. The frst ssue s onerned wth the estmaton of eah at nput port k, k and the estmaton of eah, at output port j, whh s resolved by assgnng a separate lass to every atve path. The seond ssue s related to the valdty of Eq. () for arbtrary squared oeffents of varaton and small soure rates of ndvdual paket flows. Computatonal tests reveal that, tend to onverge steadly to unty for an nreased number of routng steps. Thus, does not matter what the ntal oeffents at s the soures are, the fnal result at the destnatons an be nterpreted as a onvergene to a set of M/G/ queues wth exponental nter-arrval dstrbutons. The sad onvergene to the exponental dstrbuton, whh has maxmum entropy on the postve half-lne, s not observed n general. s ponted out by Whtt (994), for large number of lasses wth small paket flows one should expet that the resultant squared oeffents of varaton at the respetve destnatons do not dffer sgnfantly from the ntal ones at the soure. lso, Eq. () does not desrbe well the splttng of determnst paket flows (small sv) and heavy-taled ones (assumng the exstene of large sv). Whtt (994) proposed an mproved splttng formula: ρ D, μ, + ρ J j k, k k, p, k k j ( + μ, k +, (4) ( ρ ρ + ρ ), whh also takes nto aount the effet of lass-dependent serve tmes. Equaton (4) s used for the software mplementaton of a queung network analyzer n ths ontrbuton. In summary, the analyt model deals wth a frst-ome frst-serve queue and a sngle server whh an serve only one of the avalable output ports at a tme. However, dfferent output ports may have dfferent mean serve rates, μ, and squared oeffents of S, varaton,, of ther serve-tme dstrbutons. Ths requres the defnton of dfferent lasses of pakets for dfferent output ports. Knowng n advane the paths for eah soure-destnaton par, one an assoate eah paket flow wth a gven serve lass n a node. Ths a pror nformaton s needed as no lass swthng s allowed. Knowng the squared oeffents of varaton of nter-arrval and serve-tme dstrbutons as well as the server utlzaton, the omputaton of the mean queue length and the mean node delay an be performed usng the known approxmatons for GI/G/m queues. ote that queue length L and delay W are ) Regular aper 9

6 U J.T. (4): -4 (pr. 8) related by the Lttle s formula L W (Lttle 96). In order to over a wde range of squared oeffents of varaton, the followng ft for the mean watng tme s hosen (here the ndexes s, d, p,, and j go wthout sayng for the sv of eah lass of pakets) (Whtt 993): + ρ, () S W GI / G / m (,,, m) φ WM / M / m wth (Whtt 993): 4( ) φ + Ψ; 4 3 S 4 3 S φ,(6) + 3 φ3 + Ψ; ( + ) ( + ) ; + Ψ where, (7) ( ) φ 4 ; φ + γ, (8) φ 4γ, (9) ( ρ φ3 φ exp, 3ρ () φ + φ3 φ 4 mn,, () ( ρ)( m )( 4 + m ) γ mn.4,,() 6mρ where: ρ /μ, s the mean arrval rate; μ s the mean serve rate; s the sv of the nterarrval proess; and S s the sv of the servetme dstrbuton. ssume that the number of servers m. The well-known expresson for the watng tme of the M/M/ queue s provded below for ompleteness (Klenrok 97; Whtt 993; Belh et al. 998): ρ W M / M /. (3) μ( ρ) lso, n ase of a sngle server, m, whenever S and ( + ) /, t follows that φ and Eq. () redues to the llen-cunneen approxmaton (llen 99): + ρ, (4) S W GI / G / m (,,, m) WM / M / m Equaton (4) provdes an approxmaton for the watng tme when the unertanty of paket arrval s greater than the unertanty of serve. However, n wreless networks the stuaton sometmes s exatly the opposte wth S >, whh would requre the use of Eq. (). The mean node delay s obtaned as a sum of the mean watng tme and the mean serve tme: D GI / G / m WGI / G / m +. () μ The mean path delay (end-to-end delay) for a gven number of hops, h,p, s gven by: h, p D, D,,, (6) p h p h where the mean node delay n eah ntermedate node, D,p,h, s obtaned from Eq. (). The overall lateny for all paths havng a presrbed number of hops, h h, s gven by: D, h h s d h p h s d p h, p h D h h. (7) The average lateny n the network over all the atve paths for all soure-destnaton pars s gven by D s d p s d p D. Topologal Connetvty. (8) Wreless networks do not provde a full onnetvty among ther nodes at dfferent tme ntervals. Wth the nrease of the dstane between some nodes or the rtal derease of the sgnal-to-nose rato, a network s expeted to splt nto separate sub-networks wth dstnt network dameters. The operaton of the Regular aper

7 U J.T. (4): -4 (pr. 8) queung network analyzer should not be affeted by ths stuaton and the sub-networks should be evaluated for the remanng atve paket flows of the traff pattern whh an be delvered to ther respetve destnatons. The desrpton of avalable souredestnaton pars s made wth the use of the end-to-end (soure-destnaton) onnetvty matrx C, as follows:, f there s at least one path s d C., f s d or there are no paths s d (9) Then the term end-to-end onnetvty s defned as C s d x % C. (3) ( ) s d s d 3. Computatonal Examples for Sample Topologes The numeral examples nluded n ths seton demonstrate the apabltes of the sem-analyt model n evaluatng delay, throughput, utlzaton and onnetvty n a fully onneted network or n a set of subnetworks. For smplty, a set of nodes s onsdered. The nodes are ntally onneted n a x undreted mesh of equdstant nodes wth onnetvty C s d %. Eah servetme dstrbuton s desrbed by μ, pakets/se and μ,. lso, ρmx.7. Eah ntal nter-arrval dstrbuton s desrbed by pakets/se and,,,.. Shortest paths are hosen for eah soure-destnaton par. Same shortest paths are used n opposte dretons for a twoway ommunaton between arbtrary pars of nodes wthn the undreted network. Many-tomany, gateway, and unast traff patterns are tested. Fgure shows the mean end-to-end delays for dfferent path lengths for the three traff patterns. The delay of many-to-many traff s sgnfantly greater than the delays of gateway traff and unast traff due to the multple traff flows to arbtrary destnatons. The unast traff s tested on a randomly generated soure-destnaton permutaton and, for ths partular example, the maxmum path length s lmted to fve routng steps. Obvously, a mxture of the traff patterns would produe a hybrd ombnaton of delay responses. In general, the nrease of the endto-end delay wth the nrease of the number of routng steps s non-lnear and t does depend on the hosen shortest paths and the server utlzaton of eah node. The quas-lnearty shown n Fg. ours after averagng the delays of multple paths havng the same path length. The mean throughput for dfferent path lengths s shown n Fg.. o paket drop s observed for both gateway and unast patterns. The mean throughput of the many-to-many traff pattern ntally dereases but for the longest paths t tends to nrease whh an be explaned by the prnple of operaton of the teratve algorthm for rate reduton desrbed n Seton.4. n optmal algorthm produng onstant mean throughput for arbtrary path lengths s dffult to reate. lso, the rate reduton depends on the spef set of paths beng hosen for the traff pattern. The utlzaton, ρ, of the nodes s shown n Fg. 3. The many-to-many traff pattern nreases the utlzaton of some nodes to the ongeston threshold, ρ ρ MX. The gateway traff pattern tends to nrease the utlzaton of the nodes loated loser to the gateway node, whh s hosen to be at one of the orners of the mesh. End-to-End Delay, ms Many-to-Many Gateway Unast ath Length, routng steps Fg. End-to-end delay vs. path length. Regular aper

8 U J.T. (4): -4 (pr. 8) Throughput, pakets/se Utlzaton, % Many-to-Many Gateway Unast ath Length, routng steps Fg. Throughput vs. path length. Many-to-Many Gateway Unast ode Fg. 3 ode utlzaton. The unast traff pattern nreases the utlzaton at nodes wth path overlappng dependng on the hosen permutaton of soure-destnaton pars. The delvery of pakets to an arbtrary destnaton s a subjet of rate onstrants. ssumng that the mean soure rate a node generates remans the same for all nodes, several ases are tested for the many-to-many traff pattern: onstant soure rate, soure rates dereasng lnearly wth the nrease of the path length, and soure rates nreasng lnearly wth the nrease of the path length. The generaton of random soure rates for arbtrary destnatons s also onsdered. The omputatonal results are sown n Fgs There are no sgnfant dfferenes among dfferent ases for the delay shown n Fg. 4 as well as the node utlzaton shown n Fg. 6. However, the throughput shown n Fg. reflets the rate onstrants mposed on eah paket flow. For ths partular example, the mean throughput over all paths lengths for eah of the four ases s: 8. (onstant soure rate), (dereasng soure rate), 7 (nreasng soure rate), and 8 (random soure rate) pakets/se. Thus, there s no sgnfant dfferene between the mean throughput for onstant soure rates, soure rates nreasng wth the nrease of the path length, and the most ommon ombnatons of random soure rates. Ths omputatonal fnd s somewhat unexpeted as t s usually assumed that the provson of nreased soure rates for long paths nreases the overall ongeston n the network and substantally redues the throughput. The hoe of a homogeneous topology and the seleton of almost unformly overlappng paths also derease the dfferenes n the results for dfferent rate onstrants. ote that the teratve omputatons wth the queung network analyzer generate optmal results for an deal ongeston ontrol n the network. The graphs show the upper lmt of performane on the bass of an exat knowledge of the long term statsts of paket flows. ny pratal mplementaton of ongeston ontrol mehansms would produe sub-optmal results dependng on the proper estmaton of the steady state parameters at eah node. End-to-End Delay, ms Constant Soure Rate Dereasng Soure Rate Inreasng Soure Rate Random Soure Rate ath Length, routng steps Fg. 4 End-to-end delay vs. path length for dfferent rate onstrants. Regular aper

9 U J.T. (4): -4 (pr. 8) Throughput, pakets/se Constant Soure Rate Dereasng Soure Rate Inreasng Soure Rate Random Soure Rate Connetvty, % ath Length, routng steps Samplng onts Fg. Throughput vs. path length for dfferent rate onstrants. Fg. 7 Connetvty for ten onseutve samplng ponts. Utlzaton, % Constant Soure Rate Dereasng Soure Rate Inreasng Soure Rate Random Soure Rate ode End-to-End Delay, ms ath Length, routng steps S S7 S4 S Fg. 6 ode utlzaton for dfferent rate onstrants. The last example nvolves an ndoor network of slowly movng nodes. The omputatonal results are shown n Fgs. 7-. fter relatvely short transton perod followng lnk modfaton(s), one an assume the exstene of a quas-steady state for a suffent perod of tme to obtan statstal data for sub-optmal paket flow ontrol before the next hange of the network topology. Intatng ndependent random motons of the nodes, a samplng pont s reorded whenever hange(s) n the network topology our. For hosen sample transmsson ranges whh would provde mean node degrees n the range from 4 to 6 neghbors, one an reord a sequene of samplng ponts and determne the onnetvty of the quas-steady-state topology wthn a retangular area. Fg. 8 End-to-end delay vs. path length for ten onseutve samplng ponts. Throughput, pakets/se ath Length, routng steps S7 S S4 S Fg. 9 Throughput vs. path length for ten onseutve samplng ponts. Regular aper 3

10 U J.T. (4): -4 (pr. 8) Utlzaton, % S S4 S7 Samplng ont S ode Fg. ode utlzaton for ten onseutve samplng ponts. Fgure 7 shows the abrupt hanges n onnetvty for ten onseutve samplng ponts whenever the network splts nto two or more sub-networks. Fgures 8- llustrate the end-to-end delay, throughput and utlzaton for the dstnt samplng ponts. The maxmum path length vares from sample to sample as learly seen n Fgs. 8 and 9. The throughput tends to derease wth the nrease of the path length due to nonhomogeneous dstrbuton of nodes and the ourrene of topologal bottleneks. The node utlzaton hanges sgnfantly wth every samplng pont as shown n Fg. beause small hanges of the topology follow to sgnfant rearrangements of the hosen paths. 4. Conluson Ths ontrbuton ntrodues a semanalyt queung network analyzer based on ustom software wrtten n C++. lmtaton of the model s that both nter-arrval and serve-tme dstrbutons must be ndependent and have fnte mean and varane. Many-tomany traff patterns are used for the evaluaton of stat and quas-stat topologes formed by a set of ooperatve nodes. It s shown that soure rates nreasng wth the nrease of the path length are havng a mean network throughput beng smaller but lose to the one omputed for unform soure rates. The term onnetvty s ntrodued for the desrpton of sets of sub-networks whh an splt or merge at dstnt samplng ponts and the dynams of node utlzaton s observed. Referenes llen,.o. 99 robablty, statsts and queueng theory wth omputer sene applatons. adem ress rofessonal, In., San Dego, C, US. Belh, G.; Grener, S.; de Meer, H.; and Trved, K.S Queueng networks and Markov hans: Modelng and performane evaluaton wth omputer sene applatons. Wley-Intersene, John Wley & Sons, In., ew York, Y, US. ISO Open Systems Interonneton - Bas Referene Model. Klenrok, L. 97. Queueng systems, Volume : Theory, Wley Intersene, ew York, Y, US. Lttle, J.D.C. 96. roof of the Queueng Formula L W. Operatons Researh 9: 383 7, May-June. ujolle, G., and Wu, soluton for multserver and multlass open queueng networks. Informaton Systems and Operatons Researh 4(3): -3. Whtt, W The queueng network analyzer. Bell System Tehnal Journal 6(9): 779-8, ovember. Whtt, W. 983b. erformane of the queueng network analyzer. Bell System Tehnal Journal 6(9): 87-43, ovember. Whtt, W pproxmatons for the GI/G/m queue. roduton and Operatons Management (): 4-6, Sprng. Whtt, W Towards better mult-lass parametr-deomposton approxmatons for open queueng networks. nnals of Operatons Researh 48: -48. Regular aper 4