Two coupled Levy queues with independent input
aa r X i v : . [ m a t h . P R ] J un Two coupled L´evy queues withindependent input
Onno Boxma
Department of Mathematics and Computer Science, Eindhoven University of Technology,5600 MB Eindhoven, The Netherlandse-mail:
[email protected] andJevgenijs Ivanovs ∗ Department of Actuarial Science, University of Lausanne,CH-1015 Lausanne, Switzerlande-mail:
Abstract:
We consider a pair of coupled queues driven by independentspectrally-positive L´evy processes. With respect to the bi-variate workloadprocess this framework includes both the coupled processor model and thetwo-server fluid network with independent L´evy inputs. We identify thejoint transform of the stationary workload distribution in terms of Wiener-Hopf factors corresponding to two auxiliary L´evy processes with explicitLaplace exponents. We reinterpret and extend the ideas of Cohen and Boxma(1983) to provide a general and uniform result with a neat transform ex-pression.
Keywords and phrases: coupled processor model, fluid network, L´evyinput, Wiener-Hopf factorization.
1. Introduction
In the queueing literature, several studies have been devoted to a queueing modelof two servers, each with their own customer arrival process, with the specialfeature that the speed of one server changes when the other server becomes idle.This has become known as the coupled processor model. A possibly even morepopular model of two servers is a fluid network with independent arrival pro-cesses, where fixed fractions of fluid exiting one queue are routed into the sameand the other queue, as well as out of the system. These models are intimatelyrelated and in the case of L´evy input both can be put in our framework below.More specifically, we assume that our queues are driven by two indepen-dent L´evy processes X ( t ) and X ( t ) without negative jumps. We model a pairof workload processes ( W ( t ) , W ( t )) as a 2-dimensional reflected process, seee.g. Harrison and Reiman (1981); Kella (2006), W ( t ) = W (0) + X ( t ) − r L ( t ) + L ( t ) , (1) W ( t ) = W (0) + X ( t ) − r L ( t ) + L ( t ) , ∗ Supported by the Swiss National Science Foundation Project 200020-143889.1
Two coupled L´evy queues where W i ( t ) are nonnegative, L i ( t ) are nonnegative and nondecreasing with L i (0) = 0, and, in addition, it is required that if t is a point of increase of L i ( t ) then W i ( t ) = 0. Sometimes the latter condition is replaced by an equiv-alent integral condition or minimality requirement. We assume that r , r ≥ r r <
1, in which case workload processes ( W ( t ) , W ( t )) (with given ini-tial values) and unused capacity processes ( L ( t ) , L ( t )) satisfying the aboveconditions exist and are unique, see (Kella, 2006, Sec. 5). It is the easiest to understand the model given by (1) in the case of compoundPoisson inputs and constant service rates c i >
0, i.e. when each X i ( t ) is a com-pound Poisson process (CPP) minus c i t . Note that when W i ( t ) hits zero itstays at zero until the arrival of the next customer, which leads to the followingfour cases. While W ( t ) , W ( t ) > X ( t ) and X ( t ). While W ( t ) = 0 and W ( t ) > L ( t ) evolvesas c t resulting in an additional service rate r c in the second queue, i.e. thefraction r of the first server capacity is used to help the second. Similarly, while W ( t ) > , W ( t ) = 0 the service rate in the first queue is c + r c . Finally,when both queues are empty, the processes L i ( t ) evolve as certain linear driftscanceling the negative drifts of X i ( t ) and each other’s influence, which is possi-ble if and only if r r <
1. It is noted that compound Poisson input allows for aformulation of the coupled processor model, which goes beyond our assumptionof r r <
1. One simply replaces (1) by an explicit description of the workloadprocesses in the above four cases, see Cohen and Boxma (1983).As mentioned above, our model includes two-dimensional fluid networks withindependent L´evy input, where the column vector of workloads is a reflectedprocess of the form: W ( t ) = W (0) + ˜ X ( t ) − ( I − P ′ ) ct + ( I − P ′ ) ˜ L ( t ) , see e.g. Kella (1996). Here ˜ X ( t ) is a column vector of external non-decreasinginput processes into each queue, P is a routing matrix (a substochastic matrix)with P n → n → ∞ , P ′ is its transpose, I is the identity matrix, and c is a column vector of (maximal) service rates. One usually interprets ct − ˜ L ( t )as a vector of cumulative outflows from the queues, which are routed accordingto P . We can write W i ( t ) = W i (0) + X i ( t ) + (1 − p ii ) ˜ L i ( t ) − p ji ˜ L j ( t ) , where X i ( t ) = ˜ X i ( t ) − c i t + p ii c i t + p ji c j t and ( i, j ) is (1 ,
2) or (2 , L i ( t ) = (1 − p ii ) ˜ L i ( t ) and r i = p ji − p jj we obtain (1) and guarantee the aboveconditions. We remark that commonly ˜ X i ( t ) is a subordinator (a non-decreasingL´evy process) and hence X i ( t ) is a L´evy process without negative jumps hav-ing bounded variation (on finite intervals). We allow X i ( t ) to be general L´evyprocesses without negative jumps, which may lead to a certain debate about an Two coupled L´evy queues appropriate model for the fluid network, because cumulative outflows (if definedat all) are not necessarily non-decreasing in this general setup. Nevertheless, suchmodels have appeared in the literature, see Kella and Whitt (1996). Finally, onecan go the other way around and produce a network from the model (1), whichis immediate if r , r ≤
1. If r > r >
1) then consider( W ( t ) , r W ( t )) and note that it corresponds to a pair of workload processes ina network with routing matrix given by p = p = 0 , p = r r , p = 1 anddriving processes X ( t ) , r X ( t ), see also (Kella, 1996, Lem. 4.1). Let us note that ( I − P ′ ) − E X (1) <
0, where X ( t ) is a multidimensional drivingprocess, is a sufficient condition for the existence of a stationary distribution in ageneral L´evy network, which follows from Kella and Whitt (1996). Furthermore,if none of X i ( t ) is a zero process then this condition is also necessary. Strongerlimiting results are available in Kella (1996) for the case when both X i ( t ) havebounded variation. Stability of (1) can be easily related to the stability of thecorresponding network yielding the following condition d + r d > , d + r d > , (2)where d i = − E X i (1). Assuming that (2) holds we let a pair of random variables( W , W ) refer to the joint stationary distribution of ( W ( t ) , W ( t )) (uniquenessof this distribution will follow from the uniqueness of its transform). Our mainresult is an expression for the Laplace-Stieltjes transform E e − α W − α W interms of Wiener-Hopf factors corresponding to two auxiliary processes withexplicit Laplace exponents, see Theorem 1. We reinterpret and extend the ideasfrom Chapter III.3 of Cohen and Boxma (1983) to provide a general and uniformresult. Its derivation is rather compact, and is based on a number of identitiesand observations from the fluctuation theory for L´evy processes.Let us shortly discuss a special case, when X ( t ) is a subordinator, i.e. anon-decreasing L´evy process. In this case L ( t ) can increase only when L ( t )increases, hence both queues should be empty. This feature allows for a rathersimple analysis of the joint transform similarly to Kella and Whitt (1992a). Sowe can assume in the following that each X i ( t ) is a spectrally-positive L´evyprocess, i.e. it is a L´evy process which is not a subordinator, and which canhave only positive jumps. The main application/motivation of the coupled processor model is provided bythe fact that, in a network of work stations, a user may use other machines thanits own when those machines are idle; this is often referred to as cycle steal-ing . Another application occurs in integrated-service communication networks.Differentiated quality-of-service among different traffic flows is achieved in such
Two coupled L´evy queues networks via scheduling algorithms such as Weighted Fair Queueing. Mathemat-ically, such scheduling algorithms may often be represented by a form of Gen-eralized Processor Sharing , where traffic flow i gets a weight factor w i ∈ (0 , P w i = 1. If all traffic flows are backlogged, then flow i is served at rate w i .If some of the flows are not backlogged, then the excess capacity is redistributedamong the backlogged flows proportionally to their weights. Again, this may beviewed as a form of cycle stealing. A pioneering paper on the mathematical anal-ysis of coupled processors is Fayolle and Iasnogorodski (1979). They considertwo M/M/ c and c , respectively, unless the otherqueue is empty; then the speeds are c ∗ and c ∗ , respectively. They study the two-dimensional queue length process, and show how the generating function of thejoint steady-state queue length distribution can be obtained via the solution ofa Riemann-Hilbert boundary value problem. Konheim, Meilijson and Melkman(1981) provide an elegant solution of the special, slightly easier, case of two sym-metric M/M/
Two coupled L´evy queues Section 2 summarizes basic facts about spectrally-positive L´evy processes andabout Wiener-Hopf factorization of L´evy processes. In Section 3 we relate aspectrally-positive L´evy process to a certain pure-jump subordinator which playsa fundamental role in our main result, Theorem 1, formulated in Section 4.Section 5 contains the proof of Theorem 1. It basically consists of three steps. InSubsection 5.1 we derive a functional equation for the joint workload transform;in Subsection 5.2 the kernel of that functional equation is studied, and thefunctional equation is solved via Wiener-Hopf factorization assuming certainbounds; these bounds are established in Subsection 5.3. Some special cases areconsidered in Section 6, where we also discuss the result of Cohen and Boxma(1983) in the case of CPP inputs.
2. Basic facts
For ease of reference let us recall the L´evy-Khintchine formula for a spectrally-positive L´evy process X ( t ) (cf. Kyprianou (2006)): φ ( α ) = log E e − αX (1) = aα + 12 σ α − Z ∞ (1 − e − αx − αx { x< } ) ν (d x ) , (3)where ν (d x ) is a L´evy measure on (0 , ∞ ) satisfying R ∞ (1 ∧ x ) ν (d x ) < ∞ . Theprocess X ( t ) has bounded variation (on finite intervals) if and only if σ = 0 and R xν (d x ) < ∞ , in which case we have an alternative representation φ ( α ) = µα − Z ∞ (1 − e − αx ) ν (d x ) (4)and µ can be interpreted as a linear drift. The case of µ = 0 correspondsto a pure-jump subordinator. This subordinator is either a compound Poissonprocess (CPP) or an infinite activity subordinator according to ν (0 , ∞ ) beingfinite or infinite.Differentiating under the integral sign in (3), which can be justified, we get φ ′ ( α ) = a + σ α + Z x (1 − e − αx ) ν (d x ) − Z ∞ xe − αx ν (d x )for α >
0. This shows that X ( t ) has bounded variation if and only if lim α →∞ φ ′ ( α )is finite.Finally, let us recall the celebrated Wiener-Hopf factorization for a gen-eral (two-sided) L´evy process X ( t ) and some positive constant p >
0, seealso (Kyprianou, 2006, Thm. 6.16) (and comments on p. 167 about the CPPcase). Consider the Laplace transformsΨ + ( α ) = E e − αX ( e p ) , ℜ ( α ) ≥ , Ψ − ( α ) = E e − αX ( e p ) , ℜ ( α ) ≤ , (5) Two coupled L´evy queues where e p is an independent exponentially distributed r.v. with rate p and X ( t ) , X ( t )denote supremum and infimum processes respectively. Note that Ψ ± ( α ) are an-alytic in the corresponding half-planes and continuous on the imaginary axis.They satisfy the following factorization for w ∈ i R : pp − φ ( w ) = Ψ + ( w )Ψ − ( w ) . (6)Let us finally note that identification of the Wiener-Hopf factors is a difficultbut well-studied problem with some numerical evaluation techniques available,see e.g. Den Iseger, Gruntjes and Mandjes (2013).
3. Fundamental subordinators
Consider a spectrally-positive L´evy process X ( t ), which will serve as a driv-ing process in our model. The goal of this section is to associate to X ( t ) acertain pure-jump subordinator Y ( t ), which will play a fundamental role inour main result. Recall that φ ( α ) denotes the Laplace exponent of X ( t ), and d = − E X (1) = φ ′ (0) ∈ ( −∞ , ∞ ). Note that we have excluded only d = −∞ ,which is allowed because of the stability condition (2).Consider the first passage (downwards) process τ − x , x ≥
0, where τ − x =inf { t ≥ X ( t ) < − x } , which is a (possibly killed) L´evy subordinator withthe Laplace exponent − Φ( α ) defined via E e − ατ − x = e − Φ( α ) x (7)for all α with ℜ ( α ) ≥
0. For real positive α the function Φ( α ) is positive andis uniquely identified by φ (Φ( α )) = α . Moreover, lim α →∞ Φ( α ) = ∞ , and alsoΦ(0) = 0 if and only if d ≥ − α/ Φ( α ) is the Laplace exponent of a certain killed sub-ordinator (ascending ladder time process, see e.g. (Kyprianou, 2006, p. 170)).Note that if d ≥ α ↓ − α Φ( α ) = − ′ (0) = − φ ′ (0) = − d . If, however, d < d + = d ∨ Y ( t ) with the Laplace exponent φ Y ( α ) = d + − α Φ( α ) . (8)This is a pure-jump subordinator, which follows from lim α →∞ φ Y ( α ) /α = 0and representation (4). Note also that (8) holds for all α = 0 with ℜ ( α ) ≥ X ( t ):1. X ( t ) is of bounded variation: Y ( t ) is a CPP,2. X ( t ) is of unbounded variation: Y ( t ) is an infinite activity subordinator. Two coupled L´evy queues This can be seen by considering lim α →∞ φ Y ( α ) = d + − lim α →∞ ′ ( α ) = d + − lim α →∞ φ ′ ( α ), which is finite in the first case and is −∞ in the second as wasdiscussed in Section 2. So in the first case we have P ( Y (1) = 0) > P ( Y (1) = 0) = 0, which correspond to a CPP and an infinite activitysubordinator respectively.
4. Transform of the stationary workload
Consider the model specified by (1), where r i ≥ r r <
1. Recall that X ( t ) and X ( t ) are two independent spectrally-positive L´evy processes withLaplace exponents φ i ( α ), d i = − E X i (1) ∈ ( −∞ , ∞ ), and assume that stabilitycondition (2) holds. Let Y i ( t ) be a pure-jump subordinator associated to X i ( t ),whose Laplace exponent φ Yi ( α ) is given in (8). Define two L´evy processes andtwo positive constants: X L ( t ) = Y ( r t ) − Y ( t ) , X R ( t ) = Y ( t ) − Y ( r t ) , (9) p L = d +2 + r d +1 , p R = d +1 + r d +2 . Their corresponding Laplace exponents for w ∈ i R , w = 0 are given by φ L ( w ) = p R − r w Φ ( w ) + w Φ ( − w ) , φ R ( w ) = p L − w Φ ( w ) + r w Φ ( − w ) . (10)Finally, we let Ψ ± L ( α ) be the Wiener-Hopf factors corresponding to X L ( t ) andrate parameter p L . Similarly, Ψ ± R ( α ) are the Wiener-Hopf factors correspondingto X R ( t ) and p R . Theorem 1.
The joint transform of the stationary workloads is given by E e − α W − α W = 1(1 − r r )( φ ( α ) + φ ( α )) (11) × (cid:18) p R ( α − r α ) Ψ − L ( − φ ( α ))Ψ − R ( − φ ( α )) + p L ( α − r α ) Ψ + R ( φ ( α ))Ψ + L ( φ ( α )) (cid:19) , where α > Φ (0) , α > Φ (0) , and p L = d + d +1 r + d − /r , p R = d + d +2 r + d − /r . (12)It is noted that if d , d ≥ p L = p L and p R = p R . Moreover, if r i = 0then d i > d − i = 0. In this case 0 / p R , p L is interpreted as 0.Consider the above systems of queues for r = r = 0. Then X R ( t ) = Y ( t )and X R ( t ) = 0 for all t , and p R = d . From the definition of the W-H factors wehave Ψ − R ( α ) = 1 and Ψ + R ( α ) = E e − αY ( e d ) = d d − φ Y ( α ) . Plugging in α = φ ( α ) Two coupled L´evy queues we obtain Ψ + R ( φ ( α )) = d α φ ( α ) , and similarly we obtain expressions for theother terms. Putting things together we see that (11) becomes1( φ ( α ) + φ ( α )) (cid:18) d α d α φ ( α ) + d α d α φ ( α ) (cid:19) = d α φ ( α ) d α φ ( α ) , which is indeed the transform of the workload in two independent queues. An-other verification of Theorem 1 is given in Section 6.1, where we assume that X ( t ) = − d t for d >
0. For such a (degenerate) system we first provide aquick alternative derivation of the joint transform and then check it againstTheorem 1.
5. Proof
In this section we prove Theorem 1. The proof consists of three steps. In Sub-section 5.1 we derive a functional equation for the joint workload transform; inSubsection 5.2 the kernel of that functional equation is studied, and the func-tional equation is solved via Wiener-Hopf factorization assuming certain bounds;these bounds are established in Subsection 5.3.
In this section we derive an equation for the two-dimensional joint workloadtransform, which involves two unknown functions. Identification of these func-tions is the main problem, which will be addressed in Subsections 5.2 and 5.3.The following result is based on a by now standard argument using the Kella-Whitt martingale, see Kella and Whitt (1992b).
Proposition 1.
It holds that ( φ ( α ) + φ ( α )) E e − α W − α W = ( α − r α ) F ( α ) + ( α − r α ) F ( α ) , (13) where α , α ≥ and F ( α ) = E ∗ Z e − αW ( t ) d L ( t ) , F ( α ) = E ∗ Z e − αW ( t ) d L ( t ) (14) and E ∗ is the expectation in stationarity, i.e. we assume that ( W (0) , W (0)) isdistributed as ( W , W ) .Proof. Fix α , α > X ( t ) = α X ( t )+ α X ( t ) and a process of bounded variation Y ( t ) = ( α − r α ) L ( t )+( α − r α ) L ( t ), so that Z ( t ) := α W ( t ) + α W ( t ) = X ( t ) + Y ( t ). Let usfirst show that E L i ( t ) < ∞ and hence the expected variation of Y ( t ) on finiteintervals is finite. Start by noting that L ( t ) ≤ − X ( t ) + r L ( t ) , L ( t ) ≤ − X ( t ) + r L ( t ) , Two coupled L´evy queues see also Kella (2006). Hence (1 − r r ) L ( t ) ≤ − X ( t ) − r X ( t ), but it is knownthat E | X i ( t ) | < ∞ . In conclusion, M t = ( φ ( α ) + φ ( α )) Z t e − Z ( s ) d s + e − Z (0) − e − Z ( t ) − Z t e − Z ( s ) d Y ( s )is a martingale for any initial distribution, see (Kella and Whitt, 1992b, Thm. 2).Considering E ∗ M = 0 we obtain( φ ( α ) + φ ( α )) E e − α W − α W =( α − r α ) E ∗ Z e − Z ( s ) d L ( s ) + ( α − r α ) E ∗ Z e − Z ( s ) d L ( s ) . Use the properties of L i ( t ) to conclude. Remark 1.
The functional equation (13) can be used to derive the followingidentity for the means: r ( d + r d ) E W + r ( d + r d ) E W = 12 ( r φ ′′ (0) + r φ ′′ (0)) . (15) One way is to put α = r α, α = α , express F ( r α ) , differentiate it at α = 0 ,and then to do the same for α = α, α = 0 . Then the above identity follows byexpressing F ′ (0) from these equations. Note also that if d , d > then the rightside of (15) is r d E V + r d E V , where V i refers to the stationary workload inqueue i considered alone. It may be an interesting exercises to prove this relationprobabilistically from the first principles at least for Poisson inputs. Start by noting that the Laplace exponent φ i ( α ) of a spectrally-positive L´evyprocess is analytic in the right half of the complex plane and is continuous onthe imaginary axis, which can be shown from (3). Hence the same is true forΦ i ( α ), which is minus a Laplace exponent according to (7). This equation alsoimplies that ℜ (Φ i ( α )) > ℜ ( α ) >
0, which is seen by sending x to ∞ , and that Φ i ( α ) = 0 for α = 0. Hence the identity φ i (Φ i ( α )) = α extendsfrom α > α with ℜ ( α ) ≥
0. Note also that the functions F i ( α ) areanalytic on { α : ℜ ( α ) > } and continuous on the imaginary axis, which followsfrom their definition and E ∗ L i (1) < ∞ . In conclusion, Equation (13) holds forall α , α with ℜ ( α ) , ℜ ( α ) ≥ α = Φ ( w )and α = Φ ( − w ), where w ∈ i R lies on the imaginary axis, and obtain(Φ ( w ) − r Φ ( − w )) F (Φ ( − w )) + (Φ ( − w ) − r Φ ( w )) F (Φ ( w )) = 0 . Assuming w = 0 we multiply this equation by w Φ ( w )Φ ( − w ) to get( − w Φ ( − w ) + r w Φ ( w ) ) F (Φ ( − w )) = ( w Φ ( w ) − r w Φ ( − w ) ) F (Φ ( w )) , (16) Two coupled L´evy queues which immediately translates into( p L − φ L ( w )) F (Φ ( − w )) = ( p R − φ R ( w )) F (Φ ( w ))according to (10). Finally, from the Wiener-Hopf factorization (6) we have p L Ψ − R ( w )Ψ − L ( w ) F (Φ ( − w )) = p R Ψ + L ( w )Ψ + R ( w ) F (Φ ( w )) , (17)which also holds for w = 0 by continuity.The left side of (17) is analytic in the left half-plane and the right side isanalytic in the right-half plane, and both are continuous and coincide on theboundary. So one is an analytic continuation of the other, see (Lang, 1999, ThmIX.1.1). Assume for a moment that the so-obtained entire function is bounded.Then by Liouville’s theorem (Lang, 1999, Thm III.7.5) it is a constant, call it C . Let us determine the constant C , by plugging w = 0 in (17). According tothe stability condition (2) at least one of d i is positive. If d > (0) = 0and hence C = p R E ∗ L (1), whereas C = p L E ∗ L (1) if d >
0. Note also thatfor a stationary system0 = − d − r E ∗ L (1) + E ∗ L (1) and 0 = − d − r E ∗ L (1) + E ∗ L (1) , which yields E ∗ L (1) = d + r d − r r , E ∗ L (1) = d + r d − r r and provides the expression for C . Furthermore, F (Φ ( − w )) = p R − r r Ψ − L ( w )Ψ − R ( w ) , ℜ ( w ) ≤ ,F (Φ ( w )) = p L − r r Ψ + R ( w )Ψ + L ( w ) , ℜ ( w ) ≥ , where p L , p R are given in (12). This can be checked by considering three sce-narios d i ≥ , d < , d < w = − φ ( α ) for α ≥ Φ (0), so thatΦ ( − w ) = α ; for the second we let w = φ ( α ) with α ≥ Φ (0). This immediatelyyields the functions F i ( α ): F ( α ) = p R − r r Ψ − L ( − φ ( α ))Ψ − R ( − φ ( α )) , F ( α ) = p L − r r Ψ + R ( φ ( α ))Ψ + L ( φ ( α )) . This together with (13) completes the proof of Theorem 1 under the assumptionthat the entire function defined by (17) is a constant.
Two coupled L´evy queues In this section we show that the entire function defined by (17) is a constant.Consider (14) and observe that F ( α ) and F ( α ) are bounded for ℜ ( α ) ≥
0. Let˜ X L ( t ) and ˜ X R ( t ) be the processes X L ( t ) and X R ( t ), see (9), in the model definedby (1) but with interchanged indices. That is, we consider the same systemwith reversed indexing. Observe that X R ( t ) = − ˜ X L ( t ) , X L ( t ) = − ˜ X R ( t ) , p R =˜ p L , p L = ˜ p R and henceΨ − R ( − w ) = ˜Ψ + L ( w ) , Ψ − L ( − w ) = ˜Ψ + R ( w ) . (18)Therefore, it is sufficient to analyze Ψ + L ( w ) / Ψ + R ( w ) , ℜ ( w ) ≥ X L ( t ):Ψ + L ( w ) = exp (cid:18) − Z ∞ Z ∞ ( e − p L t − e − p L t − wx ) 1 t P ( X L ( t ) ∈ d x )d t (cid:19) , (19)see Thm. 6.16 and comments on p. 168 in Kyprianou (2006). Observe also thatfor ℜ ( w ) ≥ | Z ∞ Z ∞ ( e − p L t − e − p L t − wx ) 1 t P ( X L ( t ) ∈ d x )d t | ≤ Z ∞ e − p L t t d t < ∞ , | Z Z ∞ e − wx (1 − e − p L t ) 1 t P ( X L ( t ) ∈ d x )d t | ≤ p L . Hence Ψ + L ( w ) / Ψ + R ( w ) is bounded by C | e A ( w ) | , where A ( w ) := (20) Z t (cid:18)Z ∞ (1 − e − wx ) P ( X R ( t ) ∈ d x ) − Z ∞ (1 − e − wx ) P ( X L ( t ) ∈ d x ) (cid:19) d t. If both X ( t ) and X ( t ) have bounded variation then X L ( t ) and X R ( t ) are CPPsand hence Z t P ( X L ( t ) > t < ∞ , Z t P ( X R ( t ) > t < ∞ . This immediately shows that A ( w ) and hence Ψ + L ( w ) / Ψ + R ( w ) are bounded. The proof in the general case is based on the following proposition.
Proposition 2.
There exists a constant C , such that | Ψ + L ( w ) / Ψ + R ( w ) | , ℜ ( w ) ≥ is bounded by C | w | for large enough | w | . In addition, Ψ + L ( w ) / Ψ + R ( w ) = o ( w ) as w → ∞ along the real numbers. Two coupled L´evy queues Proposition 2 together with (18) implies that the entire function definedby (17) is bounded by a polynomial and hence it is a polynomial itself, see (Lang,1999, Cor. III.7.4). Taking limit along the reals shows that this polynomial isjust a constant. The proof of Proposition 2 relies on the following technicallemma.
Lemma 1.
For ℜ ( z ) ≥ it holds that Z t | − e − zt | d t ≤ | z | ∨ . For positive z the bound can be replaced by | z | ∨ .Proof. For | z | > Z t | − e − zt | d t = Z | z | t | − e − tz/ | z | | d t (21) ≤ Z | tz/ | z || t d t + Z | z | t d t = 2 + 2 ln | z | . The result is immediate for | z | < Proof of Proposition 2.
For positive w the function A ( w ) defined in (20) isbounded from above by Z t E (1 − e − wY ( t ) ; X R ( t ) > t ≤ Z t (1 − e φ Y ( w ) t )d t, because X R ( t ) ≤ Y ( t ). Recall that φ Y ( w ) ≤
0, and use Lemma 1, to bound A ( w ) by 1 + ln( | φ Y ( w ) | ∨ + L ( w ) / Ψ + R ( w ) = o ( w ).For w with | w | > , ℜ ( w ) ≥ Z t E (cid:12)(cid:12)(cid:12) − e − wX R ( t ) ; X R ( t ) > (cid:12)(cid:12)(cid:12) d t ≤ Z | w | E X + R ( t/ | w | ) t d t + Z | w | t d t ≤ E Y (1) + 2 ln | w | . A similar bound (with the constant 2 r E Y (1)) can be obtained for the terminvolving X L ( t ). Assuming that E Y (1) < ∞ we have | A ( w ) | ≤ C + 4 ln | w | for | w | > C , which yields the result. If E Y (1) = ∞ then wecan easily reduce our problem to the one, where the large jumps of Y ( t ) areremoved and hence E Y (1) < ∞ . Remark 2.
Our proof does not imply that Ψ + L ( w ) / Ψ + R ( w ) is bounded (un-less both X i ( t ) have bounded variation), and hence it leaves the possibility that F ( w ) → as w → , which is equivalent to Z { W ( t )=0 } d L ( t ) = 0 P ∗ -a.s. (and similarly for the reversed indexing). Two coupled L´evy queues
6. Special cases
Suppose X ( t ) = − d t with d ≥
0, so that φ ( α ) = d α . Then X ( t ) − r L ( t )is a non-increasing process, and so W ( t ) = 0 implying L ( t ) = r L ( t ) + d t ,and therefore W ( t ) = X ( t ) − r d t + (1 − r r ) L ( t ) . But then W ( t ) is a one-dimensional reflection of X ( t ) − r d t , and so thegeneralized Pollaczek-Khinchine formula gives E e − α W − α W = ( d + r d ) α φ ( α ) + r d α . (22)Let us check if this formula coincides with the result of Theorem 1. Note that Y ( t ) = 0 and hence Ψ − L ( α ) = Ψ − R ( α ) = 1. Also Ψ + L ( α ) = p L p L − r φ Y ( α ) andΨ + R ( α ) = p R p R − φ Y ( α ) . Therefore, we get E e − α W − α W = 1(1 − r r )( φ ( α ) + d α ) × (cid:18) p R ( α − r α ) + p L ( α − r α ) p R ( p L − r ( d +1 − φ ( α ) /α ))( p R − ( d +1 − φ ( α ) /α )) p L (cid:19) , which indeed reduces to (22). Let us only check the case when d <
0. We have d +1 = 0 , p R = r d , p L = d , p R = r d + d , p L = d + d /r = p R /r , and sothe transform reduces to r d + d (1 − r r )( φ ( α ) + d α ) (cid:18) α − r α + ( α − r α ) ( d + r φ ( α ) /α ))( r d + φ ( α ) /α )) (cid:19) , which immediately yields (22). Suppose φ i ( α ) = d i α + α /
2, i.e. the driving processes are standard Brownianmotions with drifts. Then Φ i ( α ) = − d i + p d i + 2 α and so φ Yi ( α ) = d + i + αd i − p d i + 2 α = | d i | − p d i + 2 α , which corresponds to an inverse Gaussian subordinator. Hence the processes X L and X R can be seen as differences of two inverse Gaussian processes. Theirrespective Laplace exponents are given by φ L ( w ) = φ Y ( − w ) + r φ Y ( w ) = 12 (cid:18) | d | r + | d | − q d − w − r q d + 2 w (cid:19) ,φ R ( w ) = 12 (cid:18) | d | + r | d | − r q d − w − q d + 2 w (cid:19) . The final step according to Theorem 1 is to identify the Wiener-Hopf factorscorresponding to these Laplace exponents.
Two coupled L´evy queues This subsection briefly examines relation between the general result given inTheorem 1 and the result of Cohen and Boxma (1983) for CPP inputs. Assumethat customers arrive into queue i with intensity λ i and bring iid amount ofwork distributed as B i , and the server speed is s i . Suppressing the index i , theLaplace exponent of the driving process X ( t ) is φ ( α ) = αs − λ + λ E e − αB . Therefore, φ ( α ) α = s − λ E B − E e − αB α E B = s − ρ E e − αR , where ρ = λ E B and R has the stationary residual life distribution associated to B . This further leads to α Φ( α ) = s − ρ E e − ατ − R , where R is assumed to be independent of the driving process X ( t ) and henceof its first passage time τ − x . Note that τ − R has the interpretation of the lengthof the busy period in a queue driven by X ( t ) and started with workload R . Forsimplicity we assume that s − ρ > τ − R is a proper positive randomvariable, which we denote by U .Consider − w Φ ( − w ) + r w Φ ( w ) = s − ρ E e wU + r s − r ρ E − wU (23)appearing in (16). Note that this expression can be rewritten as ( s + r s )(1 − z E e − w ˜ U ), where z = ( ρ + r ρ ) / ( s + r s ) ∈ (0 ,
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