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throughout this paper , we consider a unique server addressing two parallel queues numbered @xmath51 and @xmath52 , respectively . incoming jobs enter either queue and require random service times ; the server then processes jobs according to the so - called shortest queue first ( sqf ) policy . specifically , let @xmath6 ( resp . @xmath7 ) denote the workload in queue @xmath51 ( resp . queue @xmath52 ) at a given time , including the remaining amount of work of the job possibly in service ; the server then proceeds as follows :    * queue @xmath51 ( resp . queue @xmath52 ) is served if @xmath8 , @xmath9 and @xmath10 ( resp . if @xmath11 and @xmath12 ) ; * if only one of the queues is empty , the non empty queue is served ; * if both queues are empty , the server remains idle until the next job arrival . in contrast to fixed priority disciplines where the server favors queues in some predefined order remaining unchanged in time ( e.g. , classical preemptive or non - preemptive head - of - line priority schemes ) , the sqf policy enables the server to dynamically serve queues according to their current state . the performance analysis of such a queueing discipline is motivated by the so - called sqf packet scheduling policy recently proposed to improve the quality of internet access on high speed communication links . as discussed in @xcite , sqf policy is designed to serve the shortest queue , i.e. , the queue with the least number of waiting packets ; in case of buffer overflow , packets are dropped from the longest queue . thanks to this simple policy , the scheduler consequently prioritizes constant bit rate flows associated with delay - sensitive applications such as voice and audio / video streaming with intrinsic rate constraints ; priority is thus implicitly given to smooth flows over data traffic associated with bulk transfers that sense network bandwidth by filling buffers and sending packets in bursts . in this paper , we consider the fluid version of the sqf discipline . instead of packets ( i.e. , individual jobs ) , we deal with the workload ( i.e. , the amount of fluid in each queue ) . since the fluid sqf policy considers the shortest queue in volume , that is , in terms of workload , its performance is quantitatively described by the variations of variables @xmath6 and @xmath7 . to simplify the analysis , we here suppose that the buffer capacity for both queues @xmath51 and @xmath52 is infinite . moreover , we assume that incoming jobs enter either queue according to a poisson process ; in view of the above application context , one can argue that such poisson arrivals can model traffic where sources have peak rates significantly higher than that of the output link ; such processes can hardly represent , however , the traffic variations of locally constant bit rate flows . this poisson assumption , although limited in this respect , is nevertheless envisaged here in view of its mathematical tractability and as a first step towards the consideration of more complicated arrival patterns . the above framework enables us to define the pair @xmath13 representing the workloads in the stationary regime in each queue as a continuous - state markov process in @xmath14 . in the following , we determine the probability distribution of the couple @xmath13 by studying its laplace transform . the problem can then essentially be formulated as follows . [ prob1 ] given the domain @xmath15 and analytic functions @xmath16 , @xmath17 , @xmath18 , @xmath19 , and @xmath20 in @xmath21 , determine two bivariate laplace transforms @xmath22 , @xmath23 and two univariate laplace transforms @xmath24 , @xmath25 , analytic in @xmath21 and such that equations @xmath26 for some analytic function @xmath27 , together hold for in @xmath28 . note that each condition @xmath29 or @xmath30 with @xmath31 brings the latter equations respectively to @xmath32    to the best knowledge of the authors , the mathematical analysis of the sqf policy has not been addressed in the queueing literature . some comparable queueing disciplines have nevertheless been studied :    * the _ longest queue first _ ( lqf ) symmetric policy is considered in @xcite , where the author studies the stationary distribution of the number of waiting jobs @xmath33 , @xmath34 in each queue ; reducing the analysis to a boundary value problem on the unit circle , an integral formula is provided for the generating function of the pair @xmath35 ; * the _ join the shortest 2-server queue _ ( jsq ) , where an arriving customer joins the shortest queue if the number of waiting jobs in queues are unequal , is analyzed in @xcite . the bivariate generating function for the number of waiting jobs is then determined as a meromorphic function in the whole complex plane , whose associated poles and residues are calculated recursively . while the above quoted studies address the stationary distribution of the number of jobs in each queue , we here consider the real - valued process @xmath13 of workload components whose stationary analysis requires the definition of its infinitesimal generator on the relevant functional space . besides , the laplace transform of the distribution of @xmath13 proves to be meromorphic not on the entire complex plane , but on the plane cut along some algebraic singularities ( while the solution for jsq exhibits polar singularities only ) ; as a both quantitative and qualitative consequence , the decay rate of the stationary distribution at infinity for sqf may change according to the system load from that defined by the smallest polar singularity to that defined by the smallest algebraic singularity . the organization of the paper is as follows . in section [ ma ] , a markovian analysis provides the basic equations for the stationary distribution of the coupled queues ; the functional equations verified by the relevant laplace transforms are further derived in section [ ltd ] . in section [ expi ] , we specialize the discussion to the so - called symmetric exponential case where arrival rates are identical , and where service distribution are both exponential with identical mean ; the functional equations are then specified and shown to involve a key cubic equation . specifically , * problem [ prob1 ] * for the symmetric case is shown to reduce to the following . [ prob2 ] solve the functional equation @xmath36 for function @xmath1 , where given functions @xmath2 , @xmath3 and @xmath4 are related to one branch of a key cubic polynomial equation @xmath37 . for real @xmath38 , the solution @xmath39 is written in terms of a series involving all iterates @xmath40 for @xmath41 . the analytic extension of solution @xmath42 to some domain of the complex plane is further studied in section [ sqfqueue ] ; this enables us to derive the empty queue probability along with the tail behavior of the workload distribution in each queue for the symmetric case . the latter is then compared to that of the associated preemptive head of line ( hol ) policy . concluding remarks are finally presented in section  [ cl ] . the proofs for basic functional equations as well as some technical results are deferred to the appendix for better readability . as described in the introduction , we assume that incoming jobs consecutively enter queue @xmath43 ( resp . queue @xmath44 ) according to a poisson process with mean arrival rate @xmath45 ( resp . @xmath46 ) . their respective service times are independent and identically distributed ( i.i.d . ) with probability distribution @xmath47 , @xmath48 ( resp . @xmath49 , @xmath50 ) and mean @xmath51 ( resp . mean @xmath52 ) . let @xmath53 ( resp . @xmath54 ) denote the mean load of queue @xmath43 ( resp . queue @xmath44 ) and @xmath55 denote the total load of the system . since the system is work conserving , its stability condition is @xmath56 and we assume it to hold in the rest of this paper . in this section , we first specify the evolution equations for the system and further derive its infinitesimal generator . first consider the total workload @xmath57 of the union of queues @xmath43 and @xmath44 . for any work - conserving service discipline ( such as sqf ) , the distribution of @xmath58 is independent of that discipline and equals that of the global single @xmath59 queue . the aggregate arrival process is poisson with rate @xmath60 and the i.i.d . service times have the averaged distribution @xmath61 with mean @xmath62 . the stationary probability for the server to be in idle state , in particular , equals @xmath63 let @xmath64 ( resp . @xmath65 ) be the number of job arrivals within time interval @xmath66 at queue @xmath51 ( resp . queue @xmath52 ) ; if @xmath67 ( resp . @xmath68 ) is the service time of the @xmath69-th job arriving at queue @xmath51 ( resp . @xmath52 ) , the total work brought within @xmath66 into queue @xmath51 ( resp . @xmath44 ) equals @xmath70 ( resp . @xmath71 . denoting by @xmath72 ( resp . @xmath73 ) the workload in queue @xmath43 ( resp . @xmath44 ) at time @xmath74 , define indicator functions @xmath75 and @xmath76 by @xmath77 respectively . with the above notation , the sqf policy governs workloads @xmath72 and @xmath73 according to the evolution equations @xmath78 for @xmath79 and some initial conditions @xmath80 , @xmath81 . this defines the pair @xmath82 , @xmath83 , as a markov process with state space @xmath84 ( see figure  [ fig1 ] for sample paths of process @xmath82 ) .    as a first result , integrating each equation  ( [ path ] ) over interval @xmath85 $ ] , dividing each side by @xmath74 and letting @xmath86 implies @xmath87 and @xmath88 almost surely ( along with similar limits for integrals related to @xmath7 , @xmath89 and @xmath90 ) , and equating these limits readily provides identites @xmath91 for the stationary probability that the server treats queue @xmath51 and @xmath52 , respectively ; equivalently , the latter identities read @xmath92 in the symmetric case when arrival rates are equal and service times have identical distribution , i.e. , @xmath93 and @xmath94 , the above relations give @xmath95    the discrepancy in inequalities @xmath96 and @xmath97 defining the service policy ( when both queues are non empty ) does not favor queue @xmath43 with respect to queue @xmath44 , since event @xmath98 has probability 0 ; in fact , assuming for instance @xmath99 at some time @xmath74 , we have @xmath100 if a job arrival occurs with service time of amount exactly @xmath101 , which has probability 0 for any service time distribution . the distribution of process @xmath102 does not give a positive probability to the diagonal @xmath103 in state space @xmath104 . we now address the determination of the stationary distribution function @xmath105 , @xmath106 , of the bivariate workload process @xmath102 . in order to define the class of stationary distribution @xmath107 , we further assume that    * distribution @xmath108 has a regular density @xmath109 ( resp . @xmath110 ) at any point @xmath111 such that @xmath112 ( resp . @xmath113 ) ; * distribution @xmath107 has a regular density @xmath114 ( resp . @xmath115 ) at any point @xmath116 ( resp . @xmath117 ) on the boundary @xmath118 ( resp . on the boundary @xmath119 ) . ( a real - valued function is here said to be regular if it is continuous and bounded over its definition domain . ) in the rest of this paper , assumptions * a.1 * -*a.2 * for the existence of regular densities will be confirmed by exhibiting their laplace transforms ; the uniqueness of the stationary distribution then _ a posteriori _ justifies such assumptions . an _ a priori _ justification for the existence of densities would otherwise imply the use of malliavin calculus @xcite on the poisson space . using , we have @xmath120 in the stationary regime ; following assumptions * a.1 * -*a.2 * above , we can then write @xmath121 for all @xmath122 , where @xmath123 ( resp . @xmath124 ) is the dirac distribution at point @xmath125 ( resp . at point @xmath126 ) . + let us now characterize the stationary distribution of process @xmath127 by means of its infinitesimal generator @xmath128 defined by @xmath129\ ] ] where the limit is uniform with respect to @xmath130 ( see @xcite or ( * ? ? ? * , p.  377 ) ) ; the symbol @xmath131 denotes any function for which the latter limit exists . in the following , we denote by @xmath132 the set of functions @xmath133 everywhere bounded , twice differentiable with bounded first and second derivatives in @xmath134 . further , introduce positive cones @xmath135 along with boundaries ( see figure  [ fig1 ] ) @xmath136 we can then state the following .    with the notation ( [ cones12])-([axes12 ] ) , the infinitesimal generator @xmath128 of process @xmath127 is given by @xmath137+\lambda_2 \mathbb{e}[\theta(\mathbf{u}+\mathcal{t}_2\mathbf{e}_2)-\theta(\mathbf{u } ) ] \label{gen1}\end{aligned}\ ] ] for all @xmath130 and any test function @xmath138 , where @xmath139 ( resp . @xmath140 ) denotes the generic service time of jobs arriving at queue @xmath51 ( resp . queue @xmath52 ) and with vectors @xmath141 , @xmath142 . [ generator ]    using evolution equations ( [ path ] ) and given @xmath143 , expression ( [ gen1 ] ) is easily derived from uniform estimates ( with respect to @xmath130 ) for the distribution of the number of jumps of process @xmath144 on any interval @xmath145 $ ] ( all intervening poisson processes have rates lower than @xmath60 ) and for drift rates ( when non zero , the service rate is the constant @xmath146 ) . once generator @xmath128 is determined , the stationary distribution @xmath107 of @xmath147 is known ( see @xcite or @xcite ) to satisfy @xmath148 following the prerequisites of section  [ ma ] , we now study integral equation  . since the problem is linear in unknown distribution @xmath107 , it is tractable through laplace transform techniques . let @xmath15 and its closure @xmath149 . assumptions @xmath150 in section  [ ig ] , for the existence of regular densities @xmath151 and @xmath152 with respective support @xmath153 and @xmath154 ( see equations  ( [ cones12])-([axes12 ] ) ) enable us to define their laplace transforms @xmath22 , @xmath24 by @xmath155 for @xmath156 , where @xmath157 ; using the expectation operator , definitions ( [ ff ] ) equivalently read @xmath158 , \ ; \ ; g_1(s_1 ) = \e\big[e^{-s_1u_1}\ind_{\{0=u_2<u_1\}}\big].\ ] ] the laplace transforms @xmath23 and @xmath25 of regular densities @xmath159 and @xmath160 with respective support @xmath161 and @xmath162 ( see equations  ( [ cones12])-([axes12 ] ) ) are similarly defined by @xmath163 for @xmath156 ; equivalent definitions can be similarly written in terms of the expectation operator . expression ( [ dphi ] ) for distribution @xmath164 and the above definitions then enable to define the laplace transform @xmath165 of the pair @xmath13 by @xmath166 for @xmath167 . finally , let @xmath168 ( resp . @xmath169 ) denote the laplace transform of service time @xmath139 ( resp . @xmath140 ) at queue @xmath43 ( resp . queue @xmath44 ) for @xmath170 ( resp . @xmath171 ) ; set in addition @xmath172 and @xmath173    * a ) * transforms @xmath22 , @xmath24 and @xmath23 , @xmath25 together satisfy @xmath174 for @xmath175 , where @xmath176 and @xmath177 .    * b ) * transforms @xmath22 and @xmath25 ( resp . @xmath23 , @xmath24 ) satisfy @xmath178 for @xmath175 , with @xmath179    \\ - \lambda_2\mathbb{e } \left [ e^{-s_1u_1-s_2u_2}\ind_{\{0 \leq u_2 < u_1\}}e^{-s_2\mathcal{t}_2}\ind_{\{\mathcal{t}_2 > u_1-u_2\ } } \right ] . \label{h}\end{gathered}\ ] ]    * c ) * constants @xmath180 and @xmath181 satisfy relation @xmath182 . [ resol ]    * a ) * fix @xmath183 . the test function @xmath184 , @xmath185 , belongs to @xmath132 and has derivatives @xmath186 , @xmath187 . besides , we have @xmath188 hence @xmath189 = ( b_1(s_1)-1)\theta(\mathbf{u})$ ] , and similarly @xmath190 = ( b_2(s_2)-1)\theta(\mathbf{u})$ ] . applying proposition [ generator ] , formula ( [ gen1 ] ) for @xmath191 then yields @xmath192 with @xmath193 defined in ( [ ker ] ) . integrating that expression of @xmath191 over closed quarter plane @xmath134 with respect to distribution @xmath164 and using assumptions * a.1 * -*a.2 * , relation   then gives @xmath194 with @xmath176 and @xmath177 ; using ( [ ldef ] ) finally provides ( [ fonct ] ) . * b ) * as detailed in appendix [ a2 ] , there exists a family of functions @xmath195 with @xmath196 , such that @xmath197 , @xmath198 _ and _ @xmath199 . for given @xmath200 , @xmath201 , the function @xmath202 defined by @xmath203 , @xmath130 , therefore belongs to @xmath132 and satisfies @xmath204 pointwise in @xmath134 with @xmath205 ( note that @xmath206 ) . apply then formula ( [ gen1 ] ) to regularized test function @xmath202 and integrate this expression over @xmath134 against distribution @xmath164 to define @xmath207 in view of ( [ stationary ] ) , we have @xmath208 and , provided that @xmath209 has a finite limit @xmath210 as @xmath211 , we must have @xmath212 . the detailed calculation of that limit @xmath210 ( depending on the pair @xmath213 ) is performed in appendix [ a2 ] and condition @xmath212 is shown to reduce to first equation ( [ fonctcomp ] ) . exchanging indices 1 and 2 provides second equation ( [ fonctcomp ] ) , after noting that @xmath214 changes into @xmath215 . * c ) * adding equations ( [ fonctcomp ] ) gives ( [ fonct ] ) if and only if @xmath182 holds . computing @xmath216 by letting @xmath217 in ( [ fonct ] ) readily gives @xmath218 with @xmath219 . identity ( [ pk ] ) is obviously pollaczek - khintchin formula for the transform @xmath220 of the total workload @xmath57 in the global @xmath59 queue , with i.i.d . service times having distribution @xmath221 defined by ( [ db ] ) .    [ coroltech ] let @xmath27 be defined by ( [ h ] ) . transform @xmath24 satisfies @xmath222 for @xmath183 such that @xmath30 . similarly , transform @xmath25 satisfies @xmath223 for @xmath183 such that @xmath29 . [ c.2 ]    function @xmath224 is finite for any given @xmath183 ; if @xmath30 , the product @xmath225 is therefore zero . as @xmath226 second equation ( [ fonctcomp ] ) then implies ( [ fonctg1 ] ) . relation ( [ fonctg2 ] ) is similarly derived .      in this section , we first compare the sqf system with the hol queue , where one queue has head of line ( hol ) priority over the other ; such a comparison then enables us to extend the analyticity domain of laplace transforms @xmath22 , @xmath23 and @xmath24 , @xmath25 . + let @xmath227 , @xmath228 , denote the workload in queue @xmath229 when the other queue has hol priority ; similarly , let @xmath230 denote the workload in queue @xmath229 when this queue has hol priority over the other . finally , given two real random variables @xmath231 and @xmath232 , @xmath232 is said to dominate @xmath231 in the strong order sense ( for short , @xmath233 ) if and only if @xmath234 for any positive non - decreasing measurable function @xmath235 . workload @xmath236 verifies @xmath237 for all @xmath83 . [ domstoch ]    we clearly have @xmath238 almost surely for all @xmath83 , where @xmath75 is defined by . equation  ( [ path ] ) consequently entails that @xmath239 pathwise , which implies the strong stochastic domination . similarly , we have @xmath240 almost surely for all @xmath83 and ( [ path ] ) entails @xmath241 pathwise , hence the strong stochastic domination . assume that random variable @xmath242 has an analytic laplace transform @xmath243 in the domain @xmath244 for some real @xmath245 . laplace transform @xmath22 can be analytically extended to domain @xmath246 and transform @xmath25 can be analytically extended to @xmath247 . similarly , transform @xmath23 can be analytically extended to @xmath248 and @xmath24 can be analytically extended to @xmath249 . [ extensions ]    assume first that @xmath250 and @xmath251 are real with @xmath252 ; given @xmath253 , we have @xmath254 ; using the domination property @xmath255 of proposition [ domstoch ] and the previous inequality , definition ( [ ff ] ) of @xmath22 on @xmath149 entails @xmath158 \leq \mathbb{e}\big[e^{-(s_1+s_2)\overline{u}_2}\big];\ ] ] we then deduce that @xmath22 can be analytically continued to any point @xmath213 verifying @xmath256 and @xmath257 . assuming now that @xmath258 and @xmath259 , domination property @xmath260 yields @xmath261 and definition ( [ ff ] ) of @xmath22 on @xmath149 entails in turn @xmath158 \leq \mathbb{e}\big[e^{- s_2 \overline{u}_2}\big];\ ] ] @xmath22 can therefore be analytically continued to any point @xmath213 verifying @xmath262 and @xmath263 . we conclude that @xmath22 can be analytically continued to domain @xmath264 , as claimed . writing definition ( [ ffb ] ) of @xmath25 as @xmath265 $ ] for @xmath171 , the same type of arguments as above enables us to analytically continue function @xmath25 to any point verifying @xmath266 . domains @xmath264 and @xmath267 are illustrated in figure  [ fig3 ] ( assuming @xmath268 for instance ) .    following proposition [ resol ] and corollary  [ coroltech ] , the determination of laplace transforms @xmath22 , @xmath23 , @xmath24 and @xmath25 critically depends on both the determination of auxiliary bivariate function @xmath27 generally defined in ( [ h ] ) and the solutions to equations @xmath269 and @xmath270 . the latter , however , may be very intricate to compute for general service time distributions .    to make the resolution more tractable , we will now introduce some specific assumptions . first , service times are assumed to be exponentially distributed ; this readily provides a more explicit expression for function @xmath27 .    in the case of exponentially distributed service times , we have @xmath271 where @xmath272 are analytically defined for @xmath273 . [ hexp ]    the proof of proposition [ hexp ] is deferred to appendix [ a3 ] . expression ( [ hbis ] ) consequently reduces the determination of function @xmath27 to that of two univariate functions @xmath274 and @xmath275 .    in the rest of this paper , we further assume that the poisson arrival rates and service time distributions in each queue are equal , the so - called `` symmetric ( exponential ) case '' . because of its technical complexity , the asymmetric case will be treated in a forthcoming paper @xcite . as previously motivated , we assume from now on that    * poisson arrival rates are equal , namely @xmath93 ; * service times in both queues are exponentially distributed with identical parameter @xmath276 , i.e. , @xmath277 ;    the laplace transform of the service time distribution is then @xmath278 .    by the latter symmetry assumption , queues @xmath43 and @xmath44 are now interchangeable in terms of probability distribution . definition ( [ ff ] ) of @xmath22 or @xmath23 then entails that @xmath279 for @xmath31 and we denote by @xmath280 the latter quantity ; using similar arguments , we have @xmath281 . by proposition [ resol].c , we further have @xmath282 and function @xmath283 introduced in ( [ j1j2 ] ) is simply given by @xmath284 relations   then specialize to the unique equation @xmath285 where general expression for @xmath27 now simply reduces to @xmath286 where @xmath287 ( note the symmetry between transforms @xmath22 and @xmath23 mentioned above implies that @xmath288 ) . once function @xmath27 is expressed by ( [ hsym ] ) in terms of auxiliary function @xmath1 , functional equation ( [ kernelsym ] ) gives @xmath289 in terms of both @xmath290 and @xmath1 . as univariate transform @xmath290 will be later shown to depend on function @xmath1 only , our remaining task is therefore to derive the latter function . let us first assert some extension properties for analytic functions of interest . recall from ( * ? ? ? * 3.3 ) that the laplace transform of the workload @xmath291 in queue @xmath43 when queue @xmath44 has hol priority is given by @xmath292 = \frac{2(1-\rho)s\xi^+(s)}{\lambda(1-b(s))(s-\xi^+(s))}\ ] ] for @xmath293 , where @xmath294 is the unique root of equation @xmath295 which is positive for @xmath296 . specializing definition ( [ ker ] ) for @xmath297 to the present symmetric case , equation @xmath298 readily reduces to @xmath299 its roots @xmath294 and @xmath300 are therefore given by @xmath301 where discriminant @xmath302 is positive for @xmath303\zeta^-,\zeta^+[$ ] and non positive for @xmath304 $ ] , with @xmath305 functions @xmath306 are defined for real @xmath307 $ ] . with the convention @xmath308 , we can define analytic or meromorphic extensions of these functions in the complex plane as follows . function @xmath309 ( resp . @xmath310 ) can be analytically ( resp . meromorphically ) extended to the cut plane @xmath311 $ ] . [ extendxi ]    function @xmath312 is well - defined for @xmath313\zeta^-,\zeta^+[$ ] , whereas function @xmath314 is well - defined for @xmath315\zeta^-,\zeta^+[$ ] and @xmath316 , with @xmath317 . it is easily checked that for @xmath318 belonging to the vertical line @xmath319 , we have @xmath320 and @xmath321 ( note this vertical line and the real line are the only subsets of the complex plane on which @xmath320 ) . the schwarz s reflection principle applied to function @xmath322 with respect to the vertical line @xmath323 then ensures that the function @xmath324 defined by @xmath325 for @xmath326 and @xmath327 for @xmath328 is globally analytic on the cut plane @xmath329 $ ] . let us then define functions @xmath330 and @xmath331 by @xmath332 respectively . by construction , function @xmath330 is a meromorphic extension of @xmath310 in @xmath333 $ ] with a pole at point @xmath334 , while function @xmath331 is an analytic extension of @xmath309 in @xmath335 $ ] . + for notation simplicity , we will still denote by @xmath310 and @xmath309 their respective analytic continuation @xmath330 and @xmath331 defined above .    consider now equation @xmath336 , whose unique non - zero solution is @xmath337 . as @xmath338 , it is easily verified that solution @xmath337 is associated with branch @xmath310 if @xmath339 and with branch @xmath309 if @xmath340 . define then @xmath341 ( note that @xmath342 for all @xmath343 $ ] , as easily verified from the defining expression of polynomial @xmath344 in ( [ xi+- ] ) ) .    with the above notation , laplace transform @xmath290 can be analytically extended to the half - plane @xmath345 ; function @xmath346 can be analytically extended to @xmath347 . [ extendgf0 ]    by ( [ uhol ] ) and lemma [ extendxi ] , transform @xmath348 is analytic for @xmath349 . this transform may have a pole only at any point @xmath318 such that @xmath350 . by the above discussion , we actually have a pole at @xmath351 when @xmath352 ; it is not a pole when @xmath353 but the algebraic singularity at point @xmath354 instead occurs . applying then corollary  [ extensions ] with @xmath355 , the extended analyticity domains for @xmath290 and @xmath346 follow . following definition ( [ msym ] ) and lemma [ extendgf0 ] , function @xmath1 is consequently analytic on the half - plane @xmath356 . as detailed in section [ sqfqueue ] , the final determination of function @xmath1 relies on the algebraic and analytic properties for the branches of a cubic polynomial equation .    * a ) * for given @xmath38 and @xmath357 , relations @xmath358 can be inverted in variable @xmath318 as @xmath359 respectively , where @xmath360 and @xmath361 are the two non positive roots of cubic equation @xmath37 in variable @xmath362 , with @xmath363    for @xmath38 , @xmath300 and @xmath294 are given by @xmath364 and @xmath365 . * b ) * for @xmath366 , cubic polynomial @xmath367 has three distinct real roots @xmath360 , @xmath361 and @xmath368 such that @xmath369 and @xmath370 . [ roots - r ]    * a ) * eliminating @xmath300 between first relation ( [ z - s ] ) and polynomial equation ( [ eqxi ] ) satisfied by @xmath300 , we can write @xmath371 where @xmath372 , cubic polynomial @xmath367 being defined as in ( [ cubiceq ] ) . similarly , eliminating @xmath294 between second relation ( [ z - s ] ) and equation ( [ eqxi ] ) enables us to write @xmath373 where @xmath374 with identical polynomial @xmath367 . we readily deduce , in particular , that @xmath375 , and similarly @xmath376 . * b ) * for @xmath38 , we have @xmath377 and @xmath378 since @xmath379 by the stability condition . further accounting for its values at infinity , we deduce that cubic polynomial @xmath367 has three real roots for @xmath366 ; denoting them by @xmath360 , @xmath361 and @xmath368 , the latter discussion implies the claimed inequalities . we finally verify that roots @xmath360 and @xmath361 previously characterised either in * a ) * or * b ) * actually coincide . in fact , let @xmath380 so that @xmath381 ; given the variations of the function @xmath314 for @xmath382 , @xmath318 has to be sufficiently large for @xmath381 to be positive ; this implies that we necessarily have @xmath383 where @xmath360 is the smallest root of polynomial @xmath367 . we can similarly prove that if @xmath384 , then @xmath385 where @xmath386 is the second smallest root of @xmath367 . as solutions to a polynomial equation , algebraic functions @xmath387 , @xmath388 and @xmath389 can be analytically defined in @xmath390 cut along some slits . specifically , writing @xmath391 as @xmath392 with coefficients @xmath393 , @xmath394 and @xmath395 defined by ( [ cubiceq ] ) and introducing @xmath396 any solution @xmath397 to @xmath37 can be expressed by cardano s formula @xcite as @xmath398{\frac{1}{2 } \left ( - \widetilde{q}+ \sqrt{- \frac{\delta}{27 } } \right ) } + j^n \sqrt[3]{\frac{1}{2 } \left ( - \widetilde{q } - \sqrt{- \frac{\delta}{27 } } \right ) } , \ ] ] where @xmath399 , the pair @xmath400 can take either value @xmath125 , @xmath401 or @xmath402 , and with discriminant @xmath403 defined by @xmath404 . some algebra shows that discriminant @xmath405 factorizes as @xmath406 with @xmath407 the respective analyticity domains of functions @xmath408 , @xmath409 and @xmath410 are related to the roots of discriminant @xmath405 , these roots defining the so - called ramification points for such algebraic functions .    * a ) * discriminant @xmath405 has four distinct roots , namely two real roots @xmath411-\mu,0[$ ] and @xmath412 and two complex conjugate roots @xmath413 and @xmath414 . + * b ) * algebraic functions @xmath408 , @xmath409 and @xmath410 are analytic on the cut plane @xmath415 $ ] , @xmath416\cup[\eta_3,\eta_4])$ ] and @xmath417 $ ] , respectively . [ discriminant ]    * a ) * the point @xmath418 is clearly a root of @xmath419 and it is simple since @xmath420 in view of expression ( [ smalldelta ] ) . moreover , as the coefficient of the leading term of the cubic polynomial @xmath421 is positive , as @xmath422 and @xmath423 , discriminant @xmath405 has at least another negative real root @xmath424 between @xmath334 and 0 . besides , the discriminant of @xmath421 is easily calculated as @xmath425 with @xmath426 ; as @xmath427 for @xmath428 , we have @xmath429 . it then follows from ( * ? ? ? * theorem 1.3.1 ) that cubic polynomial @xmath421 with real coefficients has only one real root , namely @xmath424 , the two others @xmath413 and @xmath414 being complex conjugates . * b ) * by considering the analytic continuation of function @xmath430 such that @xmath431 in @xmath416\cup[\eta_3,\eta_4])$ ] , formulas ( [ cardan ] ) enable us to analytically continue function @xmath408 to the cut plane @xmath415 $ ] , function @xmath409 to the cut plane @xmath416\cup[\eta_3,\eta_4])$ ] and function @xmath410 to the cut plane @xmath417 $ ] , respectively .    the graphs of functions @xmath432 and @xmath433 are illustrated in fig . [ courbexi ] on interval @xmath434 $ ] . function @xmath435 is increasing while function @xmath436 reaches its minimum at some point @xmath437 ; @xmath436 is decreasing on interval @xmath438\zeta^+,s^*[$ ] and increasing on interval @xmath438s^*,0[$ ] . recall from proposition [ roots - r ] that @xmath383 entails @xmath439 ; conversely , we have @xmath439 for @xmath440 . function @xmath387 is thus defined and regular for @xmath441\eta_1,+\infty[$ ] where @xmath442 . using similar arguments , @xmath388 is shown to be regular for @xmath443 . on the basis of the preliminary results obtained in section [ expi ] , we are now ready to provide a final solution for auxiliary function @xmath1 ( section [ rse ] ) and determine an extended analyticity domain ( section [ ss ] ) , from which all relevant probabilistic properties for the symmetric queue can be derived ( sections [ eqp ] and [ lqa ] ) . we first provide a series expansion for laplace transform @xmath290 on some real interval of its definition domain . the proposition below states the core functional equation verified by function @xmath1 . function @xmath1 defined by ( [ msym ] ) verifies the functional equation @xmath444 for @xmath38 , with @xmath445 where @xmath371 is the unique solution to equation @xmath446 . [ c1 ]    applying equation successively to points @xmath447 and @xmath448 with identical ordinate @xmath318 , we obtain @xmath449 after using expression ( [ jj ] ) for @xmath450 and formula ( [ hsym ] ) for @xmath214 ; equations ( [ msymequ ] ) hold for sufficiently large @xmath318 so that @xmath451 is positive . using the fact that @xmath371 . equating the common value of @xmath452 from ( [ msymequ ] ) and using the fact that @xmath453 gives functional equation ( [ fonctmsym ] ) .    by proposition [ roots - r].a , @xmath454 depends on the branch @xmath360 only . as @xmath455 in view of defining equation ( [ eqxi ] ) , definition ( [ defqlh ] ) for @xmath456 further gives @xmath457 , \label{hz}\ ] ] and a similar rational expression is derived from ( [ defqlh ] ) for @xmath458 in terms of @xmath360 . as a consequence , given functions @xmath2 , @xmath3 , and @xmath4 depend only on the branch @xmath408 of cubic equation @xmath37 . note also that by the notation introduced in inversion relations ( [ z - s])-([z - sinvert ] ) , @xmath456 just coincides with @xmath459 ; the mapping @xmath460 is now introduced in view of its iterated composition , as will be shown in the central result below . the laplace transform @xmath290 can be expressed as @xmath461 \label{gsym}\ ] ] for sufficiently large real @xmath318 so that @xmath462 and where @xmath463 is given by the series expansion @xmath464 with functions @xmath2 , @xmath3 and @xmath4 defined by , and @xmath465 denoting the @xmath466-th iterate of function @xmath4 . [ resolsym ]    iterating functional equation ( [ fonctmsym ] ) for @xmath380 yields @xmath467 ( the product being equal to 1 for @xmath468 ) , with remainder @xmath469 to show that @xmath470 as @xmath471 , let us fix some @xmath380 . we first prove that the sequence @xmath472 , @xmath473 , is strictly increasing and tends to @xmath474 when @xmath475 . in fact , as @xmath476 for @xmath380 by proposition [ roots - r].b , we deduce from expression ( [ hz ] ) that @xmath477 for @xmath38 and the sequence @xmath478 , @xmath473 , is thus strictly increasing . moreover , if that sequence were upper bounded , it would tend to a finite limit @xmath479 such that @xmath480 and the number @xmath481 is positive ; but using expression ( [ hz ] ) for @xmath456 , equality @xmath480 reduces to @xmath482,\ ] ] or equivalently @xmath483 and the latter would define a simultaneously positive and negative quantity , a contradiction . we thus conclude that @xmath484 when @xmath475 .    besides , we derive from definition ( [ defqlh ] ) for @xmath2 that @xmath485 , where @xmath486 by definition ( [ msym ] ) of function @xmath1 , the sequence @xmath487 , @xmath473 , is bounded since both @xmath290 and @xmath346 vanish at infinity as laplace transforms of regular densities . it follows that remainder @xmath488 is @xmath489 = o(r^k)$ ] and therefore tends to 0 as @xmath471 . the finite sum in ( [ quotientbis ] ) thus converges as @xmath471 . formula ( [ gsym ] ) for @xmath452 eventually follows from the latter expansion inserted into second equation ( [ msymequ ] ) . we now specify the smallest singularity of laplace transform @xmath290 ; to this end , we first deal with the analyticity domain of auxiliary function @xmath1 . recall by definition ( [ msym ] ) that @xmath1 is known to be analytic at least in the half - plane @xmath356 , where @xmath490 is defined by . function @xmath1 can be analytically continued to the half - plane @xmath491 ( with @xmath492 ) defined by    * @xmath493 in case @xmath494 , where we set @xmath495 ; * @xmath496 in case @xmath340 , where @xmath497 is the largest real root of discriminant @xmath405 . [ extendm ]    the proof of proposition [ extendm ] is detailed in appendix [ a4 ] . we now turn to transform @xmath290 and determine its singularities with smallest module . recall by corollary [ extensions ] that @xmath290 has no singularity in @xmath498 . [ domaing ] the singularity with smallest module of transform @xmath290 is    * for @xmath494 , a simple pole at @xmath499 with leading term @xmath500 with @xmath501 ; * for @xmath502 , an algebraic singularity at @xmath503 with leading term @xmath504 at first order in @xmath505 , where factor @xmath506 is given by @xmath507\ ] ] with constants @xmath508 , @xmath509 and where @xmath510 , @xmath511 are given in ( [ zeta1 + - ] ) .    [ gsing ] consider again the two following cases : + * a ) * if @xmath494 , write the 1st equation ( [ msymequ ] ) as @xmath512 ; \label{gsymbis}\ ] ] as @xmath513 , we have @xmath514 while @xmath515 . proposition [ extendm ] then ensures that @xmath516 is analytic at @xmath517 since @xmath518 for @xmath494 . as @xmath452 has no singularity for @xmath519 , we conclude from expression ( [ gsymbis ] ) that @xmath290 has a simple pole at @xmath520 with residue @xmath521\ ] ] where @xmath495 . differentiating formula   for @xmath294 at @xmath520 , we further calculate @xmath522 ; residue @xmath523 in leading term ( [ resgsym1 ] ) then follows ; + * b ) * if @xmath502 , let @xmath513 so that @xmath524 and @xmath525 where @xmath526 . proposition [ extendm ] then ensures that @xmath1 is analytic at @xmath527 since @xmath528 . we conclude from expression ( [ gsymbis ] ) and the latter discussion that @xmath529 is not a singularity of @xmath290 . + by definition   of @xmath294 , where @xmath344 is factorized as @xmath530 with @xmath531 , we obtain @xmath532 where @xmath508 and with constant @xmath533 . by expression ( [ gsymbis ] ) for @xmath452 , we then obtain @xmath534 \nonumber \\ & = g(\zeta^+ ) + r^+(s-\zeta^+)^{1/2 } + ... \nonumber\end{aligned}\ ] ] since @xmath535 with @xmath536 defined in ( [ resgsym2 ] ) . expansion ( [ resgsym2 ] ) then follows with associated factor @xmath506 ; we conclude that the singularity with smallest module of @xmath290 is @xmath510 , an algebraic singularity with order 1 . the results obtained in the previous section enable us to give a closed - form expression for the empty queue probability in terms of auxiliary function @xmath1 only . probability @xmath537 is given by @xmath538 with @xmath1 given by series expansion ( [ mseries ] ) . [ g0sym ]    apply relation ( [ gsymbis ] ) for @xmath452 with @xmath539 ; as @xmath540 , we then derive that @xmath541 hence @xmath542 ; \label{g(0)bis}\ ] ] differentiating formula   for @xmath294 at @xmath539 gives @xmath543 so that the first term inside brackets in ( [ g(0)bis ] ) reduces to @xmath544 . now , applying ( [ fonctmsym ] ) to value @xmath539 ( with corresponding pair @xmath545 and @xmath546 ) shows that the right - hand side of ( [ g(0)bis ] ) also equals @xmath547 , as claimed .    by ( [ p0 ] ) and ( [ g(0 ) ] ) , we derive the probability @xmath548 that either queue @xmath43 or @xmath44 is empty . we depict in figure  [ figg0 ] the variations of @xmath549 in terms of load @xmath550 when fixing @xmath551 ( for comparison , the black dashed line represents the empty queue probability @xmath552 for the unique queue aggregating all jobs from either class @xmath43 or @xmath44 ) . the numerical results show that @xmath549 decreases to a positive limit , approximately @xmath553 , when @xmath554 tends to 1 ; this can be interpreted by saying that , while the global system is unstable and sees excursions of either variable @xmath6 or @xmath7 to large values , one of the queues remains less than the other for a large period of time and has therefore a positive probability to be emptied by the server . furthermore , the red dashed line depicts the empty queue probability @xmath555 if the server were to apply a preemptive hol policy with highest priority given to queue @xmath43 ; following lower bound ( [ stochlowup ] ) , we have @xmath556 . we further notice that for @xmath557 , the positive limit of @xmath549 derived above for sqf is close enough to the maximal limit @xmath558 of @xmath559 . the above observations consequently show that the sqf policy compares favorably to the optimal hol policy by guaranteeing a non vanishing empty queue probability for each traffic class at high load . we finally derive asymptotics for the distribution of workload @xmath6 or @xmath7 in either queue , i.e. , the estimates of tail probabilities @xmath560 for large queue content @xmath561 . we shall invoke the following tauberian theorem relating the singularities of a laplace transform to the asymptotic behavior of its inverse ( * ? ? ? * theorem 25.2 , p.237 ) . let @xmath165 be a laplace transform and @xmath562 be its singularity with smallest module , with @xmath563 as @xmath564 for @xmath565 and @xmath566 ( replace @xmath165 by @xmath567 if @xmath568 is finite ) . the laplace inverse @xmath235 of @xmath165 is then estimated by @xmath569 for @xmath570 , where @xmath571 denotes euler s @xmath571 function . [ tauberian ]    note that the fact that @xmath568 is finite or not does not change the estimate of inverse @xmath235 at infinity . before using that theorem for the tail behavior of either @xmath6 or @xmath7 , we first state some simple bounds for their distribution tail .    the global workload @xmath57 is identical to that in an @xmath572 queue with arrival rate @xmath573 and service rate @xmath276 . the complementary distribution function of @xmath58 is therefore given by @xmath574 for all @xmath575 , with @xmath495 ; the distribution tail of workload @xmath6 or @xmath7 therefore decreases at least exponentially fast at infinity . following upper bound ( [ stochlowup ] ) relating @xmath6 to variable @xmath291 corresponding to a hol service policy with highest priority given to queue @xmath44 , we further have @xmath576 for all @xmath577 . the laplace transform of @xmath291 is given by equation   and is meromorphic in the cut plane @xmath311 $ ] , with a possible pole at @xmath578 . specifically , the application of theorem [ tauberian ] shows that the tail behavior of @xmath291 is given by @xmath579 for large @xmath561 . the tail behavior of @xmath291 , and therefore @xmath6 , may therefore be either exponential or subexponential according to system parameters . we precisely have the following result . the workload in queue @xmath43 is such that @xmath580 for large @xmath561 , with constants @xmath495 and @xmath581 where @xmath582 is given by ( [ zeta1 + - ] ) and @xmath506 by ( [ resgsym2 ] ) . [ asymptoticssym ]    applying equation  ( [ ldef ] ) to @xmath583 gives the laplace transform of @xmath6 as @xmath584 with @xmath585 by using and ( [ hsym ] ) . we now follow the results of theorem [ gsing ] on the smallest singularity of @xmath290 in order to derive the smallest singularity of transform @xmath586 expressed above . @xmath587 assume first @xmath494 . by proposition [ extendm ] , function @xmath588 is analytic for @xmath589 . it then follows from ( [ laplu1 ] ) that the singularity with smallest module of @xmath590 is at @xmath591 with leading term @xmath592 since @xmath593 and the root @xmath594 of @xmath595 is a removable singularity since @xmath596 has to be analytic for @xmath597 . by estimate ( [ resgsym1 ] ) for @xmath598 near @xmath591 , ( [ leadterm0 ] ) yields @xmath599 as @xmath600 ; smallest singularity @xmath591 is thus a simple pole for laplace transform @xmath586 . applying then theorem [ tauberian ] with @xmath601 and @xmath602 , we derive that @xmath603 for large @xmath561 with prefactor @xmath604 as claimed . @xmath587 assume now that @xmath502 . by formula ( [ laplu1 ] ) and proposition [ extendm ] , function @xmath588 is analytic for @xmath605 . it then follows from ( [ laplu1 ] ) that the singularity with smallest module of @xmath590 is at @xmath606 with leading term again specified by ( [ leadterm0 ] ) so that @xmath607 \label{smallestpolegter}\ ] ] near @xmath606 . by estimate ( [ resgsym2 ] ) , ( [ smallestpolegter ] ) yields @xmath608 as @xmath609 where @xmath610 smallest singularity @xmath606 is thus an algebraic singularity for laplace transform @xmath586 , with order @xmath611 . applying theorem [ tauberian ] with @xmath612 , @xmath613 and @xmath614 , we derive that @xmath615 for large @xmath561 with prefactor @xmath616 . @xmath587 finally , assume that @xmath617 ; the polar singularity @xmath618 and the algebraic singularity @xmath619 for @xmath290 coincide in this case . recall from proposition [ extendm].b that function @xmath620 is analytic for @xmath621 whenever @xmath340 ; @xmath424 is the only real zero @xmath622 of discriminant @xmath406 and expression ( [ smalldelta ] ) of @xmath421 gives @xmath623 , hence @xmath624 ; @xmath516 is therefore analytic at @xmath625 . near @xmath626 , formula ( [ xi+- ] ) easily gives @xmath627 expression ( [ gsymbis ] ) for @xmath598 and the discussion above then imply that @xmath628 in the neighborhood of @xmath626 . the leading term ( [ leadterm0 ] ) for @xmath590 is consequently given by @xmath629 smallest singularity @xmath626 is thus an algebraic singularity for laplace transform @xmath586 , with order @xmath630 . applying then theorem [ tauberian ] with @xmath631 , @xmath632 and @xmath633 , we derive that @xmath634 for large @xmath561 with prefactor @xmath635 .    for any given load @xmath6360,1[$ ] , theorem [ asymptoticssym ] consequently provides the same exponential trend as that of upper bound ( [ asympthol ] ) for hol ; as a matter of fact , a large value of @xmath6 entails that queue @xmath43 behaves as if queue @xmath44 , with smaller workload , had a hol priority . the stationary analysis of two coupled queues addressed by a unique server running the sqf discipline has been generally considered for poisson arrival processes and general service time distributions ; required functional equations for the derivation of the stationary distribution for the coupled workload process have been derived . specializing the resolution of such equations to both exponentially distributed service times and the so - called `` symmetric case '' , all quantities of interest have been obtained by solving a single functional equation .    the solution @xmath1 for that equation has been given , in particular , as a series expansion involving all consecutive iterates of an algebraic function @xmath4 related to a branch of some cubic equation @xmath37 . it must be noted that the curve represented by that cubic equation in the @xmath637 plane is singular ; in fact , whereas `` most '' cubic curves are regular ( i.e. , without multiple points ) , it can be easily checked that cubic @xmath638 has a double point at infinity . in equivalent geometric terms , cubic @xmath638 can be identified with a sphere when seen as a surface in @xmath639 , whereas most cubic curves are identified with a torus . this fact can be considered as an essential underlying feature characterizing the complexity of the present problem ; such geometric statements will be enlightened for solving the general asymmetric case in @xcite . an extended analyticity domain for solution @xmath1 has been determined as the half - plane @xmath640 , thus enabling to determine the singularity of laplace transform @xmath290 with smallest module . it could be also of interest to compare such extended domain @xmath640 to the maximal convergence domain of series expansion ( [ mseries ] ) ( recall the convergence of that series has been stated in theorem [ resolsym ] for real @xmath38 only ) ; in fact , the analyticity domain @xmath640 may not coincide with the validity domain for such a series representation . the discrete holomorphic dynamical system defined by the iterates @xmath641 , @xmath642 , definitely plays a central role for such a comparison . as an alternative approach to that of section [ sqfqueue ] , function @xmath1 may also be derived through a riemann - hilbert boundary value problem ; hints for such an approach can be summarized as follows . we successively note that    * there exists @xmath643\zeta^-,\zeta^+[$ ] such that for @xmath644 , @xmath645 belongs to the analyticity domain @xmath640 determined by proposition  [ extendm ] ; * denoting by @xmath646 the image by functions @xmath647 of the open interval @xmath438s_0,\zeta^+[$ ] , we note that @xmath648 for @xmath649 with @xmath650 . equations   then enable us to deduce the condition @xmath651    the above riemann - hilbert problem for function @xmath1 is , however , valid on open path @xmath646 only and not on the whole closed contour @xmath652 , defined as the image by functions @xmath653 of closed segment @xmath654 $ ] . the well - posed problem , nevertheless , formulates as follows . [ rhpglob ] determine a function @xmath107 which is analytic in @xmath655 , where @xmath656 is the domain delineated by the closed contour @xmath652 , tends to 0 at infinity and such that boundary condition ( [ bv0 ] ) holds on @xmath652 ( and not only on @xmath646 ) . if the solution @xmath107 to * problem 3 * can be shown to exist and to be analytic on @xmath656 , then functions @xmath1 and @xmath107 coincide . proving the latter statement and deriving an alternative representation of solution @xmath1 ( namely , as a path integral on closed contour @xmath652 ) is an object of further study .    on the application side , the performance of the sqf discipline has been characterized , both in terms of empty queue probability and distribution tail at infinity . the results show that sqf compares quite favorably with respect to the `` optimal '' priority discipline , namely hol . such performance properties will be generalized to the asymmetric case where flow patterns are allowed to be heterogeneous . before proving equations ( [ fonctcomp ] ) , we state preliminary expressions of @xmath24 and @xmath25 .    given @xmath657 univariate transforms @xmath24 and @xmath25 satisfy @xmath658 [ g1g2prov ]    as transforms of regular densities , we have @xmath659 , @xmath660 when @xmath661 for fixed @xmath250 with @xmath200 . besides , we have @xmath662 , @xmath663 when @xmath661 with fixed @xmath664 , where @xmath665 is the laplace transform of the restriction of density @xmath159 on the boundary @xmath154 and @xmath181 is the value at @xmath666 of density @xmath160 on boundary @xmath162 ; as a consequence , @xmath667 for fixed @xmath664 . now , letting @xmath251 tend to @xmath474 in each side of ( [ fonct ] ) , the above limit results entail @xmath668 with @xmath669 , which provides identity ( [ g1g2comp ] ) for @xmath670 . identity ( [ g1g2comp ] ) for @xmath671 is symmetrically deduced by letting @xmath250 tend to @xmath474 in ( [ fonctcomp ] ) with fixed @xmath672 . we now address the derivation of equations ( [ fonctcomp ] ) . recall that subsets @xmath153 , @xmath154 , etc . of state space @xmath134 are defined in ( [ cones12])-([axes12 ] ) . given @xmath196 , define the function @xmath673 by @xmath674 ; @xmath673 is twice continuously differentiable over @xmath675 , @xmath676 for each @xmath677 and @xmath678 ( the dirac mass at @xmath679 ) for the weak convergence of distributions . for given @xmath200 , @xmath201 , let then be the test function @xmath203 , @xmath130 , with @xmath680 function @xmath202 belongs to @xmath681 and is 0 on the outside of @xmath153 ; moreover , we have @xmath198 so that @xmath204 pointwise in @xmath134 , with limit function @xmath131 defined by @xmath682 , @xmath130 .    by direct differentiation , we further calculate @xmath683 for @xmath130 , with @xmath684 after ( [ iepsilon ] ) ; note that derivative @xmath685 tends to @xmath686 for the weak convergence of distributions as @xmath211 . + let us now calculate the limit @xmath687 with @xmath209 introduced in ( [ mepsilon ] ) ; to this end , we address successive terms of @xmath688 according to definition ( [ gen1 ] ) . integrating first @xmath689 over @xmath690 against @xmath164 reduces to @xmath691 \varphi_1(\mathbf{u})\mathrm{d}\mathbf{u}\ ] ] since @xmath689 vanishes on the outside of @xmath153 ; on account of the above mentioned weak convergence properties , we then obtain @xmath692 with @xmath693 defined as in lemma [ g1g2prov ] and where @xmath694 defines the laplace transform of density @xmath151 restricted to the positive diagonal @xmath695 ( function @xmath696 is determined below ) . besides , the integral of @xmath697 over @xmath698 equals 0 as this function vanishes on the outside of @xmath153 . further , we have @xmath699 for given @xmath130 and @xmath700 , therefore @xmath701 by the dominated convergence theorem ; hence @xmath702 where random variable @xmath127 has distribution @xmath107 . for given @xmath130 , we similarly have @xmath703 and therefore @xmath704 finally , noting that @xmath705 and adding limit terms ( [ lim1 ] ) , ( [ lim3 ] ) , ( [ lim4 ] ) according to ( [ gen1 ] ) gives limit @xmath706 the final expression @xmath707   \\ & + \lambda_1 \left [ b_1(s_1)g_2(s_2 ) - \int_0^{+\infty } e^{-s_2u_2}\mathbb{e } \left ( e^{-s_1\mathcal{t}_1 } \ind_{\{\mathcal{t}_1 > u_2\ } } \right ) \psi_2(u_2)\mathrm{d}u_2 \right ]   \\ & + \lambda_2 b_2(s_2 ) f_1(s_1,s_2 )   + \lambda_2 \ ; \bigg [ \int_{\gamma_2 } e^{-\mathbf{s } \cdot \mathbf{u } } \mathbb{e } \left ( e^{-s_2\mathcal{t}_2 } \ind_{\{\mathcal{t}_2 > u_1 - u_2\ } } \right ) \varphi_2(\mathbf{u } ) \mathrm{d}\mathbf{u }    \\ & +   \int_0^{+\infty } e^{-s_1u_1}\mathbb{e } \left ( e^{-s_2\mathcal{t}_2 } \ind_{\{\mathcal{t}_2 > u_1\ } } \right   ) \psi_1(u_1)\mathrm{d}u_1 \bigg ] - \lambda f_1(s_1,s_2 ) = 0.\end{aligned}\ ] ]    defining @xmath214 as in ( [ h ] ) to gather all remaining integrals , the latter identity reads @xmath708 with @xmath709 , @xmath693 being defined by ( [ defe21e12 ] ) and where @xmath696 defines the laplace transform of density @xmath151 restricted to the diagonal . changing index 1 into 2 , and noting that @xmath214 changes into @xmath710 , symmetrically yields second equation @xmath711 with @xmath712 , @xmath713 defined by and where @xmath714 defines the laplace transform of density @xmath159 restricted to the diagonal . to conclude the proof , we prove the following technical lemma .    functions @xmath696 and @xmath714 are identically zero . [ w1w2 ]    adding equations ( [ fonctcomp1 ] ) and ( [ fonctcomp2 ] ) ( and omitting arguments for the sake of simplicity ) yields @xmath715 . on the other hand , equation ( [ fonct ] ) gives @xmath716 ; equating right hand sides of the latter equations then provides the identity @xmath717 using expressions ( [ g1g2comp ] ) for @xmath670 and @xmath671 , the latter identity simply reduces to @xmath718 , showing that function @xmath719 is constant . as both @xmath696 and @xmath714 vanish at @xmath474 , this constant is 0 and since these functions are non negative by definition , this entails that @xmath720 .    after using equations ( [ g1g2comp ] ) to express @xmath693 and @xmath713 in terms of @xmath671 and @xmath670 , respectively , lemma [ w1w2 ] finally enables us to reduce ( [ fonctcomp1 ] ) and ( [ fonctcomp2 ] ) to equations ( [ fonctcomp ] ) . this concludes the proof of proposition [ resol ] . for an exponentially distributed service time @xmath139 with parameter @xmath721 , the factor of @xmath45 in definition ( [ h ] ) of @xmath214 reads @xmath722 = \\ \int_0^{+\infty } \int_0^{+\infty } \left [ \int_{u_2-u_1}^{+\infty } e^{-s_1x_1 } \mu_1 e^{-\mu_1x_1 } \mathrm{d}x_1 \right ] e^{-s_1u_1-s_2u_2}\ind_{\{0 \leq u_1 < u_2\}}\mathrm{d}\phi(u_1,u_2),\end{gathered}\ ] ] for @xmath723 and @xmath724 . by definition ( [ dphi ] ) , the latter term is equal to @xmath725   \\   = \frac{\mu_1}{\mu_1+s_1 } \big [ g_2(s_1+s_2+\mu_1 ) + f_1(-\mu_1,s_1+s_2+\mu_1 ) \big ] \end{gathered}\ ] ] where , by corollary [ extensions ] , each term inside brackets is analytically defined for @xmath213 such that @xmath726 and @xmath727 , respectively , that is at least for @xmath728 . similarly , for an exponentially distributed service time @xmath140 with parameter @xmath729 , the factor of @xmath46 in definition ( [ h ] ) of @xmath214 reads @xmath730 =   \\ - \frac{\mu_2}{\mu_2+s_2}\left [ g_1(s_1+s_2+\mu_2 ) + f_2(s_1+s_2+\mu_2,-\mu_2)\right ] \end{gathered}\ ] ] where , by corollary [ extensions ] , each term inside brackets is analytically defined for @xmath213 such that @xmath731 and @xmath732 , respectively , hence for @xmath733 . adding up the two above expressions , we obtain claimed expressions . * by lemma [ discriminant ] , function @xmath387 is analytic on the cut plane @xmath736 $ ] , where ramification points @xmath737 , @xmath424 are determined as the real negative roots of discriminant @xmath405 . as @xmath738 , function @xmath387 is , in particular , analytic in the half - plane @xmath739 ; * by definition ( [ cubiceq ] ) , we may have @xmath740 only if @xmath741 , that is , @xmath742 or @xmath743 or @xmath744 ; in the case @xmath745 , we have @xmath746 and in the case @xmath747 , @xmath748 we conclude that we can not have @xmath749 if @xmath750 ; * by corollary [ extensions ] , transform @xmath290 is analytic on @xmath751 where @xmath752 if @xmath494 and @xmath354 if @xmath502 . * a ) * assume first that @xmath352 . in the @xmath756 plane , the diagonal @xmath757 intersects the curve @xmath758 at @xmath759 ( see fig .  [ courbexi1 ] ) . further , we easily verify that @xmath760 for @xmath761 and condition ( [ conditionaa ] ) is therefore fulfilled in this first case . we then conclude that function @xmath1 is analytic for @xmath762 , and thus for @xmath763 ( recall by definition ( [ msym ] ) that @xmath1 is the sum of two non - negative laplace transforms ) . * b ) * assume now that @xmath340 ( see fig .  [ courbexi2 ] ) . we have shown above that we can not have @xmath764 , which would otherwise imply @xmath765 . we thus necessarily have @xmath766 , which entails that @xmath767 for @xmath768 and condition ( [ conditionaa ] ) is therefore fulfilled in this second case . we then conclude that function @xmath1 is analytic for @xmath768 , hence for @xmath621 .
we analyze the so - called shortest queue first ( sqf ) queueing discipline whereby a unique server addresses queues in parallel by serving at any time that queue with the smallest workload . considering a stationary system composed of two parallel queues and assuming poisson arrivals and general service time distributions , we first establish the functional equations satisfied by the laplace transforms of the workloads in each queue . 
 we further specialize these equations to the so - called `` symmetric case '' , with same arrival rates and identical exponential service time distributions at each queue ; we then obtain a functional equation @xmath0 for unknown function @xmath1 , where given functions @xmath2 , @xmath3 and @xmath4 are related to one branch of a cubic polynomial equation . 
 we study the analyticity domain of function @xmath1 and express it by a series expansion involving all iterates of function @xmath4 . 
 this allows us to determine empty queue probabilities along with the tail of the workload distribution in each queue . 
 this tail appears to be identical to that of the head - of - line preemptive priority system , which is the key feature desired for the sqf discipline .