On the excursions of reflected local time processes and stochastic fluid queues
aa r X i v : . [ m a t h . P R ] J un (25 October 2018) ON THE EXCURSIONS OF REFLECTED LOCAL TIMEPROCESSES AND STOCHASTIC FLUID QUEUES
TAKIS KONSTANTOPOULOS, ∗ Heriot-Watt University
ANDREAS E. KYPRIANOU, ∗∗ University of Bath
PAAVO SALMINEN, ∗∗∗ ˚Abo Akademi University
Abstract
This paper extends previous work by the authors. We consider the local timeprocess of a strong Markov process, add negative drift, and reflect it `a laSkorokhod. The resulting process is used to model a fluid queue. We derive anexpression for the joint law of the duration of an excursion, the maximum valueof the process on it, and the time distance between successive excursions. Wework with a properly constructed stationary version of the process. Examplesare also given in the paper.
Keywords:
L´evy process, local time, Skorokhod reflection, stationary process2000 Mathematics Subject Classification: Primary 60G51, 60G10Secondary 90B15
1. Introduction
Consider a stationary strong Markov process X = ( X t , t ∈ R ), defined onsome filtered probability space (Ω , F , P, ( F t , t ∈ R )), with values in R + , a.s.c`adl`ag paths, and adapted to ( F t ). In this paper, the local time L of theprocess X at x = 0 is considered an ( F t )-adapted stationary random measurethat regenerates jointly with X at every (stopping) time that X hits 0. Moreprecisely: ∗ Postal address: School of Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS,UK ∗∗ Postal address: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath,BA2 7AY, UK ∗∗∗
Postal address: Department of Mathematics, ˚Abo Akademi University, Turku, FIN-20500, Finland (A1) L assigns a nonnegative random variable L ( B, ω ) to each B ∈ B ( R ) suchthat L ( · , ω ) is a Radon measure for each ω ∈ Ω . (A2) For any a.s. finite ( F t ) -stopping time T at which X T = 0 , the process (( X T + t , L ( T, T + t )) , t ≥ is independent of F T . We take the broader perspective with regard to the process L and we allowfor the case that it is a local time of an irregular point (in which case L hasdiscontinuous paths) as well as the case that 0 is a sticky point (in which case L is absolutely continuous with respect to the Lebesgue measure with density c ( X t = 0) for some c > § V.3] for further discussion. For each s ∈ R define theinverse local time process with respect to t by L − s ; u := inf { t > L [ s, s + t ] > u } , u ≥ . (1)What is important is that, owing to this definition, the inverse of the cumulativelocal time is a L´evy process in the following sense: Lemma 1. If L is continuous then for every a.s. finite ( F t ) -stopping time T such that X T = 0 , the process ( L − T ; u , u ≥ is a subordinator with L − T ;0 = 0 . If L is not continuous, that is to say if 0 is an irregular point for X , then thisLemma is taken as an additional requirement to the definition of L . This iseasily arranged by choosing L to be a modification of the counting process on Z , the discrete set of times that X visits 0, so that the inverse is a subordinator.To do this, we assign, to each element of Z , an i.i.d., unit-mean exponentiallydistributed weight. Then let the local time on an interval I to be the sum ofall the weights of the points of Z in I .We summarise this as an assumption, in addition to (A1)-(A2) above: (A3) If L is discontinuous then we require that for every a.s. finite ( F t ) -stoppingtime T such that X T = 0 , the process ( L − T ; u , u ≥ is a subordinator. We will also need the following assumption: n the excursions of reflected local time processes (A4) The stationary random measure L has finite rate not exceeding , i.e. EL (0 , t ) = µ t, where < µ < . Then, as in [19], [16], [22], and [15], we define a stationary process Q =( Q t , t ∈ R ) by Q t = sup −∞
2. A closer look at the reflected process
Consider now any a.s. finite ( F t )-stopping time T , such that X T = 0. Then( L − T ; t , t ≥
0) is a subordinator starting from zero (owing to Lemma 1 orassumption (A3)) with law that does not depend on T . It turns out thatthe process of interest is Λ T = { Λ T,t : t ≥ } , whereΛ T ; t = t − L − T ; t , t ≥ . (4)Note that, irrespective of T , the process Λ T has the law of the same boundedvariation, spectrally negative L´evy process which is issued from the origin attime zero. By (A4), L has rate µ <
1; hence E Λ T ;1 = 1 − µ <
0. Since L − T ; t is a subordinator, it has a well-defined, possibly nonzero, drift. If this drift islarger than or equal to unity then − Λ T is a subordinator and, as it will turnout, this is a trivial case.We therefore assume in the sequel that the drift of L − T is less than unity or,equivalently, that (A5) The drift δ Λ of the process Λ defined by (4) is strictly positive. Under this assumption, the point 0 is irregular for ( −∞ ,
0) for Λ T (thisfollows as a standard results for bounded variation spectrally negative L´evyprocesses, see Bertoin [2, Chap. VII].)In addition, under (A5), it is clear that the time taken for Λ T to first enter( −∞ ,
0) is almost surely strictly positive. It will be shown below (Lemma3) that this implies that the excursions of the process Q , i.e. the busy periods,have strictly positive Lebesgue length with probability one. It can be intuitivelyseen, via a geometric argument involving the reflection of the space-time path n the excursions of reflected local time processes Figure The construction of the process (Λ T ; t , t ≥ and related processes, assuming that T = 0 . Note that Λ may have countably many jumps on finite intervals. of Λ T about the diagonal (see Figure 1), that the time taken for Λ T to firstenter ( −∞ ,
0) is almost surely equal to the length of the excursion of Q startedat time T .In this light, note also that Λ T cannot creep downwards because it is spec-trally negative with paths of bounded variation (cf. Bertoin [2, Chap. VII]).Hence the overshoot at first passage of Λ T into ( −∞ ,
0) is almost surely strictlypositive. It will turn out (Lemma 1) that this overshoot agrees with the idleperiod following the aforementioned excursion of Q .The above analysis implies that, on finite intervals of time, Q has finitelymany excursions (busy periods) separated by positive-length idle periods. De-note by · · · < g ( − < g (0) < g (1) < g (2) < · · · the beginnings of the idle periods and by · · · < d ( − < d (0) < d (1) < d (2) < · · · Takis Konstantopoulos, Andreas E. Kyprianou, Paavo Salminen their ends, see Figure 2. We choose the indexing so that g (0) ≤ < g (1). Let N g (respectively, N d ) be the point process with points { g ( n ) : n ∈ Z } (respectively, { d ( n ) : n ∈ Z } ). As Q is a stationary process, N g and N d are jointly stationarywith finite, nonzero, intensity [15] denoted by λ (an expression for which isgiven by (26) and is derived in § g (1) g (0) * (−1) (0) d d (1) d IB Q sample path of Q Figure The definition of g ( n ) and d ( n ) . By convention, the origin of time is between g (0) and g (1) , under the original measure P . Under P d , the origin of time is at d (0) . Under P g ,the origin of time is at g (0) . The random variable Q ∗ is the maximum deviation from of Q within the typical busy period. N g , N d we have the Palm probabilities P g , P d , respectively. Let us consider Q under the measure P d . Then P d ( d (0) = 0) = 1, i.e. the origin of time is placedat the beginning of a busy period. By the strong Markov property, the “cycles” C n := { Q t : d ( n ) ≤ t < d ( n + 1) } , n ∈ Z , are i.i.d. under measure P d . In particular, the pairs of random variables (cid:0) g ( n + 1) − d ( n ) , d ( n + 1) − g ( n + 1) (cid:1) , n ∈ Z , are i.i.d. under P d . Consider the triple( B, I, Q ∗ ) := g (1) − d (0) , d (1) − g (1) , sup d (0) Lemma 2. Let D = inf { t > X t = 0 } and d = inf { t > Q t > } . Then d = D a.s. on { Q = 0 } . We now obtain an alternative expression for B = g (1) − d (0) and I = d (1) − g (1) in terms of the inverse local time. Lemma 3. We have that B = g (1) − d (0) = inf { u > L − d (0); u > u } , (6) B + I = d (1) − d (0) = L − d (0); g (1) − d (0) . (7) Proof. Since d (0) is the end of an idle period (and the beginning of a busyperiod), we have Q d (0) − = 0. Using then expression (3) we obtain Q t = L [ d (0) , t ] − ( t − d (0)) , d (0) ≤ t < g (1) , which gives B = g (1) − d (0) = inf { t > L [ d (0) , d (0) + t ] = t } . Consider now L − d (0); u , defined by (1). By Lemma 2, d (0) is a point of increase ofthe function t L [ d (0) , d (0) + t ]. Hence g (1) > d (0). Also, when L [ d (0) , d (0) + t ] − t decreases, it does so continuously. Therefore, B = inf { t > L [ d (0) , d (0) + t ] < t } . Notice also that, for all t, x > L [ d (0) , d (0) + t ] < x ⇐⇒ t < L − d (0); x − , where L − d (0); x − = lim ε ↓ L − d (0); x − ε . It follows that, B = inf { t > t < L − d (0); t − } = inf { t > L − d (0); t > t } , Takis Konstantopoulos, Andreas E. Kyprianou, Paavo Salminen by the right continuity of t L − d (0); t . To prove the expression for B + I , noticethat L does not charge the interval [ g (1) , d (1)) because, by definition, Q is zerofor all t in this interval. (cid:3) Henceforth it will be convenient to work with the process Λ = (Λ t , t ≥ t := t − L − d (0); t , t ≥ . Note also that d (0) is an ( F t )-stopping time at which X takes the value 0 andhence in our earlier notation Λ t = Λ d (0); t .From the expression (6), and as discussed in the introduction of Section 2,we see that B is simply the first time at which Λ enters ( −∞ , B = inf { t > t < } , (8)which is necessarily strictly positive thanks to the irregularity of 0 for ( −∞ , I = L − d (0); g (1) − B = L − d (0); B − B = − Λ B , (9)i.e. I is, in absolute value, equal to the value of Λ at the first time it becomesnegative. Again we recall from the discussion at the beginning of Section 2 thatΛ cannot creep downwards and hence I > Q ∗ = sup d (0) 3. The triple law Recall that P d is the Palm probability with respect to the point process { d ( n ) , n ∈ Z } . The function H ( α, β, x ) = E d h e − αB − βI ( Q ∗ ≤ x ) i n the excursions of reflected local time processes characterizes the joint law of the triple ( B, I, Q ∗ ) under P d . Since P d ( d (0) =0) = 1, we have that Λ t = t − L − t , with Λ = 0, P d -a.s. (12)Recalling the expressions (8), (9) and (11) for B , I and Q ∗ , respectively, wewrite H ( α, β, x ) = E d h e − αB + β Λ B ( B ≤ τ x ) i . (13)Since our primary object is the process Λ defined in (12), and in view of (4)and (13), it makes sense to consider the process on its canonical probabilityspace and denote its law by P . Then H ( α, β, x ) = E h e − αB + β Λ B ( B < τ x ) i . (14)The latter function may now be expressed in terms of so-called scale functionsfor spectrally negative L´evy processes. To define the latter, let ψ Λ ( θ ) := log E e θ Λ , θ ≥ , be the Laplace exponent of Λ under P . Then the, so-called, q -scale functionfor (Λ , P ), denoted by W ( q ) ( x ), satisfies W ( q ) ( x ) = 0 for x < , ∞ ) itis the unique continuous (right continuous at the origin) monotone increasingfunction whose Laplace transform is given by Z ∞ e − θx W ( q ) ( x ) dx = 1 ψ Λ ( θ ) − q , for β > Φ Λ ( q ) , (15)where Φ Λ ( q ) = sup { θ ≥ ψ Λ ( θ ) = q } is the right inverse of ψ Λ . (See for example the discussion in Chapter 9 of [17]). Theorem 1. Let Λ be the process defined by (12) , B its first entry time to ( −∞ , as in (8) , and τ x the first hitting time of { x } as in (10) . For α, β, x ≥ we have H ( α, β, x ) = E h e − αB + β Λ B ( B < τ x ) i = 1 − δ Λ α − ψ Λ ( β )) R x e − βy W ( α ) ( y ) dye − βx W ( α ) ( x ) . (16) Proof. Let G t := σ (Λ s , s ≤ t ) and define, for all β ≥ 0, the exponential( G t )-martingale M βt := e β Λ t − ψ Λ ( β ) t , t ≥ . Let, on the canonical space of Λ, P β be a probability measure, absolutelycontinuous with respect to P on G t for each t , with Radon-Nikod´ym derivative d P β d P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G t := M βt . Notice that Λ is still a L´evy process under P β with Laplace exponent ψ β Λ ( θ ) = log E β e θ Λ = ψ Λ ( β + θ ) − ψ Λ ( β ) . (17)It is straightforward to check from the above formula that, under P β , Λ isspectrally negative, with bounded variation paths and drift coefficient equal to δ Λ . Since on the stopped σ -field G B we have ( d P β /d P ) (cid:12)(cid:12) G B = M βB , we maysubstitute e β Λ B = M βB e ψ Λ ( β ) B in the equation (14) for H to obtain H ( α, β, x ) = E h M βB e ψ Λ ( β ) B e − αB ( B < τ x ) i = E β h e − ( α − ψ Λ ( β )) B ( B < τ x ) i . Let q := α − ψ Λ ( β ) , and assume that q ≥ 0. It follows from [17, Thm, 8.1(iii)] that H ( α, β, x ) = E β (cid:2) e − qB ( B < τ x ) (cid:3) = Z ( q ) β (0) − Z ( q ) β ( x ) W ( q ) β (0) W ( q ) β ( x ) , (18) n the excursions of reflected local time processes where W ( q ) β is the q -scale function for (Λ , P β ) and Z ( q ) β is given by Z ( q ) β ( x ) = 1 + q Z x W ( q ) β ( t ) dt. It is easy to see [17, Lemma 8.4] that the Laplace transform of W ( q ) β ( · ) is theLaplace transform of W ( q ) ( · ) shifted by β and this ensures that W ( q ) β ( x ) = e − βx W ( α ) ( x ) . (19)Moreover, since Λ still has drift coefficient δ Λ under P β , [17, Lemma 8.6] tells usthat, irrespective of the value of q and β , W ( q ) β (0) = 1 /δ Λ . Putting the piecestogether, this gives us the desired expression for α ≥ ψ Λ ( β ). However [17,Lemma 8.3], since W ( q ) ( x ) is analytic in q , the condition on α can be relaxedto α ≥ (cid:3) In view of (8), (9), (11), and (13), we get the following corollary. Corollary 1. (Joint law of typical B , I and Q ∗ .) Assume that (A1)–(A5) hold.Then the joint law of the length B of a typical busy period, the length I of atypical idle period, and the maximum Q ∗ of Q over the typical busy period isexpressed by the formula E d [ e − αB − βI ( Q ∗ ≤ x )] = 1 − δ Λ α − ψ Λ ( β )) R x e − βy W ( α ) ( y ) dye − βx W ( α ) ( x ) (20) where α, β, x ≥ . 4. Marginal distributions Clearly, formula (20) can be used to extract more detailed information abouttypical behaviour of Q . Let us first derive the distribution (Laplace transform)of the pair ( B, I ) under the measure P d . We have E d [ e − αB e − βI ] = E [ e − αB e β Λ B ] = lim x →∞ H ( α, β, x ) . To derive the limit, let us temporarily assume that q = α − ψ Λ ( β ) > β ≥ 0. Consider (16) in the form (18) and use the limiting resultlim x →∞ Z ( q ) β ( x ) W ( q ) β ( x ) = q Φ β Λ ( q ) , from [17, Exercise 8.5] where the function Φ β Λ is the right inverse of ψ β Λ . Thatis to say Φ β Λ ( q ) = sup { θ ≥ ψ β Λ ( θ ) = q } = sup { θ ≥ ψ β Λ ( θ ) = α − ψ Λ ( β ) } = sup { θ ≥ ψ Λ ( θ + β ) = α } = Φ Λ ( α ) − β. This gives E d [ e − αB e − βI ] = 1 − δ Λ α − ψ Λ ( β )Φ Λ ( α ) − β . (21)To remove the restriction that α > ψ Λ ( β ) in (21) and replace it instead by just α ≥ 0, one may again proceed with an argument involving analytical extensiontaking care to note for the case that α = ψ Λ ( β ),lim | α − ψ Λ ( β ) |→ α − ψ Λ ( β )Φ Λ ( α ) − β = lim | α − ψ Λ ( β ) |→ ψ β Λ (Φ Λ ( α ) − β )Φ Λ ( α ) − β = ψ β Λ ′ (0+) = ψ ′ Λ ( β ) . Letting β = 0 in (21), we find the P d -law of B . That is to say, E d [ e − αB ] = 1 − δ Λ α Φ Λ ( α ) . This formula is consistent with the result of [15, Prop. 8] and, moreover, we seethat the mean duration of the busy period is given by E d [ B ] = 1 δ Λ Φ Λ (0) . (22) n the excursions of reflected local time processes To find the P d -law of I we need to set α = 0. Recall however from thebeginning of Section 2 that E d (Λ ) < 0. This implies that Φ Λ (0) > E d [ e − βI ] = 1 − δ Λ ψ Λ ( β ) β − Φ Λ (0) . (23)It follows that the mean idle period is thus equal to E d [ I ] = − ψ ′ Λ (0+) δ Λ Φ Λ (0) , (24)where ψ ′ Λ (0+) = E d (Λ ) < A cycle of the process Q is defined as the interval from the beginning of abusy period until the beginning of the next busy period. We therefore havemean cycle length = E d [ B + I ] = 1 − ψ ′ Λ (0+) δ Λ Φ Λ (0) . (25)We can express the common rate, λ , of N g and N d as the inverse of the meancycle length: λ := EN d (0 , 1) = EN g (0 , 1) = 1 E d [ B + I ] = δ Λ Φ Λ (0)1 − ψ ′ Λ (0+) . (26) We now derive the P d -distribution of Q ∗ . Letting α = β = 0 in (16) weobtain P d ( Q ∗ ≤ x ) = 1 − δ Λ W ( x ) , where W ( x ) ≡ W (0) ( x ) is defined through its Laplace transform Z ∞ e − θx W ( x ) dx = 1 ψ Λ ( θ ) , for θ > Φ Λ (0) . (27)An immediate observation is that lim x → P d ( Q ∗ ≤ x ) = 0, since W (0) =lim θ → θ/ψ Λ ( θ ) = 1 /δ Λ . So under P d , the random variable Q ∗ has no atomat zero–which is, of course, expected. We now show that Q ∗ has exponential tail under P d and derive the preciseasymptotics. To do this, let β ∗ := Φ Λ (0) . Then (27) gives that the Laplace transform of x e − β ∗ x W ( x ) is θ /ψ Λ ( β ∗ + θ ). From the final value theorem for Laplace transforms,lim x →∞ e − β ∗ x W ( x ) = lim θ → θψ Λ ( β ∗ + θ ) = 1 ψ Λ ′ ( β ∗ ) , where we used the fact that ψ Λ ( β ∗ ) = 0. It follows that P d ( Q ∗ > x ) ∼ ψ ′ Λ (Φ Λ (0)) δ Λ e − Φ Λ (0) x as x → ∞ . 5. Cycle formulae We now show how the use of cycle formulae of Palm calculus enable us tofind (Proposition 1 below) the joint law of the endpoints of an idle periodconditional on the event that the idle period contains the origin of time. Also(Proposition 2 below) we characterise the joint law of the endpoints of a busyperiod, together with the maximum of Q over this busy period, conditional onthe event that the busy period contains the origin of time.It is well-known that if (Ω , F , P ) is endowed with a P -preserving flow ( θ t , t ≥ 0) (see end of Section 1) then for any random measure M with finite intensity λ M , and any point process N with finite intensity λ N such that M ( B, θ t ω ) = M ( B + t, ω ), N ( B, θ t ω ) = N ( B + t, ω ), for t ∈ R , B Borel subset of R , and ω ∈ Ω, and any nonnegative measurable Z : Ω → R , we have λ M E M [ Z ] = λ N E N Z T k +1 T k Z ◦ θ t M ( dt ) , (28)where P M , E M (respectively, P N , E N ) denotes Palm probability and expecta-tion with respect to M (respectively, N ), T is the first atom of N which is ≤ 0, and T k , T k +1 are any two successive atoms of N . n the excursions of reflected local time processes The next result can be found for some special cases in [16] (diffusions), [15](L´evy processes), and in [22] the general expression is derived. Here we offer anew proof in the general case based on (28). Proposition 1. (Joint law of endpoints of idle period.) Assume that (A1)–(A5) hold. Then, conditional on Q = 0 , the left end-point, g (0) , and rightend-point, d (0) , of the idle period containing t = 0 have joint Laplace transformgiven by E [ e − αd (0)+ βg (0) | Q = 0] = Φ Λ (0) − ψ ′ Λ (0+) · α − β (cid:18) ψ Λ ( α ) α − Φ Λ (0) − ψ Λ ( β ) β − Φ Λ (0) (cid:19) , for non-negative α and β ( α = β ).Proof. Let M I be the restriction of the Lebesgue measure on the idle periods: M I ( A ) = Z A ( Q t = 0) dt, A ∈ B ( R ) . Then E M I [ Z ] = E [ Z | Q = 0] for all nonnegative random variables Z . Apply(28) with M = M I , N = N d , and Z = e − αd (0)+ βd (0) : λ M I E M I [ e − αd (0)+ βg (0) ] = λE d Z d (0) d ( − e − αd (0) ◦ θ t + βg (0) ◦ θ t M I ( dt ) . Here λ is the rate of N d and is given by (26). The rate λ M I is given by λ M I = E d [ I ] E d [ B + I ] . Hence λλ M I = 1 E d [ I ] = δ Λ Φ Λ (0) − ψ ′ Λ (0+) , where we used (24) and (25). Now, P d ( d (0) = 0) = 1. To compute the integralabove, note that M I is zero on the interval ( d ( − , g (0)), and that, for g (0) ≤ t ≤ 0, we have d (0) ◦ θ t = − t , and g (0) ◦ θ t = g (0) − t . So the integral aboveequals Z g (0) e ( α − β ) t − αg (1) dt = e βg (0) − e αg (0) α − β . Combining the above we obtain E [ e − αd (0)+ βg (0) | Q = 0] = Φ Λ (0) − ψ ′ Λ (0+) · E d [ e βg (0) ] − E d [ e αg (0) ] α − β . Since E d [ e βg (0) ] = E d [ e − βI ], the result is obtained by using (23). (cid:3) Proposition 2. (Joint law of endpoints of busy period and maximum over it.) Assume that (A1)–(A5) hold. Then, conditional on Q > , the left end-point, d (0) , and right end-point, g (1) , of the busy period containing t = 0 , togetherwith the maximum of Q s for s ranging over this busy period have a joint lawwhich is characterised by E [ e − αg (1)+ βd (0) ( Q ∗ ≤ x ) | Q > Λ (0) α − β (cid:18) α R x W ( α ) ( y ) dyW ( α ) ( x ) − β R x W ( β ) ( y ) dyW ( β ) ( x ) (cid:19) , (29) for non-negative α and β ( α = β ).Proof. Let M B be the restriction of the Lebesgue measure on the busyperiods: M B ( A ) = Z A ( Q t > dt, A ∈ B ( R ) . Then E M B [ Z ] = E [ Z | Q > 0] for all random variables Z ≥ 0. Apply (28): λ M B E M B [ e − αg (1)+ βd (0) ( Q ∗ ≤ x )]= λE d Z d (1) d (0) e − αg (1) ◦ θ t + βd (0) ◦ θ t ( Q ∗ ◦ θ t ≤ x ) M B ( dt )= λE d Z g (1)0 e − α ( g (1) − t ) − βt ( Q ∗ ≤ x ) dt = λE d (cid:20) ( Q ∗ ≤ x ) e − αg (1) e ( α − β ) g (1) − α − β (cid:21) = λα − β (cid:18) E d [ e − βg (1) ( Q ∗ ≤ x )] − E d [ e − αg (1) ( Q ∗ ≤ x )] (cid:19) = λα − β ( H ( β, , x ) − H ( α, , x )) , (30) n the excursions of reflected local time processes where H ( α, β, x ) is the right-hand side of (20). Using (26), (25) and (22), wehave λλ M B = 1 E d [ B ] = δ Λ Φ Λ (0) . Combining the above we obtain the announced formula. (cid:3) Proposition 2 yields the next corollary which recovers a result obtained in [22]using different methods (for special cases, see [16] and [15]). Clearly, Corollary2 could also be proved analogously as Proposition 1. Corollary 2. Assume that (A1)–(A5) hold. Then, conditional on Q > , theleft end-point, d (0) , and right end-point, g (1) , of the busy period containing t = 0 have joint Laplace transform given by E [ e − αg (1)+ βd (0) | Q > 0] = Φ Λ (0) α − β · (cid:18) α Φ Λ ( α ) − β Φ Λ ( β ) (cid:19) , for non-negative α and β ( α = β ).Proof. The argument proceeds as in the proof Proposition 2 by omittingthe factor ( Q ∗ ≤ x ), i.e. by formally replacing x with + ∞ . The last line of(30) will give λα − β ( H ( β, , ∞ ) − H ( α, , ∞ )), where H ( α, β, ∞ ) is given by theright-hand side of (21). (cid:3) Corollary 3. Assume that (A1)–(A5) hold. Then, conditional on Q > , themaximum of Q over the busy period containing t = 0 has distribution P ( Q ∗ ≤ x | Q > 0) = Φ Λ (0) W ( x ) R x W ( y ) dy − R x W ( x − y ) W ( y ) dyW ( x ) (31) for x ≥ .Proof. Letting α, β → P ( Q ∗ ≤ x | Q > 0) = Φ Λ (0) lim α → ∂ ˆ H∂α ( α, , x ) , where ˆ H ( α, , x ) = 1 − α R x W ( α ) ( y ) dyW ( α ) ( x ) . Next recall that for each x > W ( α ) ( x ) is an entire function in the variable α and in particular W ( α ) ( x ) = X k ≥ α k W ∗ ( k +1) ( x )where W ∗ ( k +1) ( x ) is the ( k + 1)-th convolution of W (cf. Bertoin [3]). Fromthis one easily deduces that ∂∂α W ( α ) ( x ) | α =0 = Z x W ( y ) W ( x − y ) dy. The result now follows from straightforward differentiation. (cid:3) 6. Example: Local time storage from reflected Brownian motion withnegative drift Let X = { X t , t ∈ R } be a reflected Brownian motion with drift − c < I = [0 , ∞ ) , and let P denote the probability measureassociated with X when initiated from 0 at time 0. Its local time (at 0) for s < t is given by L ( s, t ] := lim ε ց ε Z ts [0 ,ε ) ( X u ) du. (32)Let Q be the stationary process defined as in (2): Q t := sup s ≤ t { L ( s, t ] − ( t − s ) } . This particular example of fluid queues was introduced and analysed in [19]and further studied in [16] and [15].Recall that E L (0 , 1] = c, and, hence Q is well-defined if and only if 0 1. Here we make this example more complete by finding the α -scale functionassociated with process Λ t := t − L − t , t ≥ , where L − t := inf { s : L (0 , s ] > t } , t ≥ . is the inverse local time process. As seen from formulae (31) and (16), the α -scale function is the key ingredient needed for computing the distribution ofthe maximum of Q over a busy period and related random variables. n the excursions of reflected local time processes To begin with, we recall some basic formulae. When normalising as in (32),see [10, pg. 214], [6, pg. 22] it holds that E (exp {− θL − t } ) = exp (cid:26) − t Z ∞ (1 − e − θu ) 1 √ πu e − c u/ du (cid:27) = exp (cid:26) − tG θ (0 , (cid:27) , (33)where G θ (0 , 0) := 1 √ θ + c − c is the resolvent kernel (Green kernel) of X at (0 , E (exp { θ Λ t } ) = exp (cid:26) t (cid:18) θ − G θ (0 , (cid:19)(cid:27) = exp { tψ Λ ( θ ) } , where ψ Λ ( θ ) := θ − p θ + c + c, θ ≥ . Recall (cf. (15)) that the α -scale function ( α ≥ 0) associated with Λ isdefined for x ≥ Z ∞ e − θx W ( α ) ( x ) dx = 1 ψ Λ ( θ ) − α ; (34)for x < W ( α ) ( x ) = 0 . The 0-scale function is called simply the scalefunction and denoted W. For the next proposition introduceErfc( x ) := 2 √ π Z ∞ x e − t dt, and notice that Erfc(0) = 1 , Erfc(+ ∞ ) = 0 , and Erfc( −∞ ) = 2 . Proposition 3. The α -scale function W ( α ) of Λ is for x ≥ given by W ( α ) ( x ) = e − c x/ λ − λ (cid:16) λ e λ x/ Erfc( − λ p x/ − λ e λ x/ Erfc( − λ p x/ (cid:17) , (35) where λ := 1 + p (1 − c ) + 2 α, λ := 1 − p (1 − c ) + 2 α (36) In particular, W ( x ) = e − c x/ − c ) (cid:16) (2 − c ) e (2 − c ) x/ Erfc( − (2 − c ) p x/ − c e c x/ Erfc( − c p x/ (cid:17) , (37) and W (0) = 1 . Proof. From (34) we have Z ∞ e − θx W ( α ) ( x ) dx = 1 θ − √ θ + c + c − α . (38)To invert this Laplace transform, introduce λ := 2 θ + c . With this notation,1 θ − √ θ + c + c − α = 2 λ − √ λ + 2( c − α ) − c = 2( √ λ − λ )( √ λ − λ ) (39)= 2 λ − λ (cid:18) √ λ − λ − √ λ − λ (cid:19) , where λ , are the roots of the equation z − z + 2( c − α ) − c = 0 , i.e., as in(36). Next, recall the following Laplace inversion formula (cf. Erd´elyi [7, pg.233]) L − (cid:18) √ λ + β (cid:19) = 1 √ πx − β e β x Erfc( β √ x ) (40)valid for λ − β > . Since Z ∞ e − θx W ( α ) ( x ) dx = Z ∞ e − λy e c y W ( α ) (2 y )2 dy we obtain using (40)2e c y W ( α ) (2 y ) = 2 λ − λ (cid:16) λ e λ y Erfc( − λ √ y ) − λ e λ y Erfc( − λ √ y ) (cid:17) , which is formula (35). In particular, when α = 0 it holds λ = 2 − c and λ = c yielding formula (37). (cid:3) Using the scale function W and the factΦ Λ (0) = sup { θ > ψ Λ ( θ ) = 0 } = 2(1 − c ) n the excursions of reflected local time processes formula (31) yields the distribution of the maximum Q ∗ over an observed busyperiod (i.e. over a busy period containing the origin of time). Proposition 4. Let < c < . The distribution of the maximum Q ∗ over anobserved busy period of a reflected Brownian motion (with drift − c ) local timestorage is given by P ( Q ∗ ≤ x | Q > 0) = 2(1 − c ) R x W ( y ) ( W ( x ) − W ( x − y )) dyW ( x ) , (41) where the scale function W is given by (37). We plot the derivative of (41) for c = 1 / Figure The density of Q ∗ conditional on { Q > } for the example corresponding toBrownian motion with drift − c = − / . We recall some formulas from [16]. First E (cid:16) e θd (0) − βg (1) | Q > (cid:17) = 8(1 − c ) p θ + (1 − c ) + p β + (1 − c ) × p θ + (1 − c ) + 1 + c )( p β + (1 − c ) + 1 + c )=: F ( θ, β ; 1 − c ) , (42)and E (cid:16) e θg (0) − βd (0) | Q = 0 (cid:17) = F ( θ, β ; c ) . (43) Setting β = θ in the right-hand side of (42) and (43), respectively, we get E (cid:16) e − θ ( g (1) − d (0)) | Q > (cid:17) = 4(1 − c ) p θ + (1 − c ) ( p θ + (1 − c ) + 1 + c ) , (44)and E (cid:16) e − θ ( d (0) − g (0)) | Q = 0 (cid:17) = 4 c √ θ + c ( √ θ + c + 2 − c ) , (45)Taking the inverse Laplace transform of (44) (cf. Erd´elyi [7, pg. 234]) we obtainthe density of the length of the busy period g (1) − d (0), given Q > 0, as f g − b ( v ) = 2(1 − c )e − (1 − c ) v/ (cid:16)p v/π − (1 + c ) ve (1+ c ) v/ Erfc((1 + c ) p v/ (cid:17) Note that the density of the length of the idle period d (0) − g (0), given that Q = 0, is obtained from f g − b ( v ) by substituting c for 1 − c . In Figure 4 wehave ploted f g − b ( v ) for three different values of c . We notice also that the mean Figure The density of length of the busy period, given that Q > for three different valuesof c . busy period length has a simple expression: E [ g (1) − d (0) | Q > 0] = 2 − c (1 − c ) . n the excursions of reflected local time processes The joint density of d (0) and g (1) is given by P ( − d (0) ∈ dx, g (1) ∈ dy | Q > 0) = 2(1 − c )e − (1 − c ) ( x + y ) / × (cid:16)p / ( π ( x + y )) − (1 + c ) e (1+ c ) ( x + y ) / Erfc((1 + c ) p ( x + y ) / (cid:17) and, again, the density for ( g (0) , d (0)) is obtained by substituting c for 1 − c. Next we find the density of g (1) (recall that − d (0) is identical in law with g (1)) by inverting the Laplace transform (obtained from (42) by choosing θ =0): E (cid:16) e − βg (1) | Q > (cid:17) = 4(1 − c ) (cid:16)p β + (1 − c ) + 1 − c (cid:17) (cid:16)p β + (1 − c ) + 1 + c (cid:17) . (46)Letting λ := 2 β + (1 − c ) we rewrite (46) as E (cid:16) e − βg (1) | Q > (cid:17) = 2(1 − c ) c (cid:18) √ λ + 1 − c − √ λ + 1 + c (cid:19) . From (40) L − (cid:18) √ λ + 1 − c − √ λ + 1 + c (cid:19) = (1 + c ) e (1+ c ) x Erfc (cid:0) (1 + c ) √ x (cid:1) − (1 − c ) e (1 − c ) x Erfc (cid:0) (1 − c ) √ x (cid:1) . (47)Consequently,2e − (1 − c ) x f g (1) (2 x ) = 2(1 − c ) c (cid:16) (1 + c ) e (1+ c ) x Erfc((1 + c ) √ x ) − (1 − c ) e (1 − c ) x Erfc((1 − c ) √ x ) (cid:17) , (48)where f g (1) denotes the density of g (1) conditioned on { Q > } . From (48) weobtain f g (1) ( x ) = (1 − c ) e − (1 − c ) x/ c (cid:16) (1 + c ) e (1+ c ) x/ Erfc (cid:16) (1 + c ) p x/ (cid:17) − (1 − c ) e (1 − c ) x/ Erfc (cid:16) (1 − c ) p x/ (cid:17) (cid:17) . (49) Moreover, the density f d (0) of d (0) conditional on { Q = 0 } is obtained from(49) by substituting c instead of 1 − c : f d (0) ( x ) = c e − c x/ − c (cid:16) (2 − c ) e (2 − c ) x/ Erfc (cid:16) (2 − c ) p x/ (cid:17) − c e c x/ Erfc (cid:16) c p x/ (cid:17) (cid:17) . (50)It is striking how similar formulae (37) and (50) are. Remark 1. The scale function formulae (35) and (37) are clearly valid for all c ≥ . In case c = 0 the process { L (0 , t ] ; t ≥ } is a version of the Brownianlocal time, and the α -scale function W ( α )0 of the corresponding process Λ isgiven by W ( α )0 ( x ) = e (1+ α ) x √ α (cid:16) (1 + √ α ) e x √ α Erfc (cid:16) − (1 + √ α ) p x/ (cid:17) − (1 − √ α ) e − x √ α Erfc (cid:16) − (1 − √ α ) p x/ (cid:17) (cid:17) . In particular, W ( x ) = e x Erfc( −√ x ) . (51)In case c = 1 it holds λ , = 1 ± √ α and for α = 0 formula (37) can be useddirectly. For the 0-scale function we need to take the limit as c → W ( x ) = (1 + x ) Erfc( − p x/ 2) + r xπ e − x/ . (52) Remark 2. Here we display some formulae for Laplace transforms apparentfrom above and point out a misprint in Erd´elyi et al. [7].First, from (38), (39), and (52) we have the following Laplace inversionformula valid for λ > L − (cid:18) √ λ − (cid:19) = (1 + 2 x ) e x Erfc( −√ x ) + 2 √ x √ π , (53)and this can be “extended” (for a > 0) to L − (cid:18) √ λ − a ) (cid:19) = (1 + 2 a x ) e a x Erfc( − a √ x ) + 2 a √ x √ π . (54) n the excursions of reflected local time processes Furthermore, it can be checked that (54) is valid for all a < L (cid:0) √ x (cid:1) = √ π λ − / , L (cid:16) e a x Erfc( a √ x ) (cid:17) = λ − / ( λ / + a ) − , and L (cid:16) x e a x Erfc( a √ x ) (cid:17) = − ∂∂λ L (cid:16) e a x Erfc( a √ x ) (cid:17) = − ∂∂λ λ − / ( λ / + a ) − = 12 a (cid:16) λ − / − λ − / ( λ / + a ) − (cid:17) . We remark that formula (10) in [7] p. 234: L − (cid:18) √ λ + √ b ) (cid:19) = 1 − p bx/π + (1 − bx ) e bx (cid:16) Erf( √ bx ) − (cid:17) . (55)is not correct since it does not coincide with formula (54) (for a < x ) := 2 √ π Z x e − t dt, the right-hand side of (55) is zero at zero but the right-hand side of (54) is 1at zero. 7. Further examples In the previous example we derived a local time process from a given Markovprocess. However, it is also possible to consider examples where just the localtime process L , or equivalently the subordinator L − , is specified. Indeed thesubordinator that will play the role of L − in this example has no drift and hasL´evy measure given byΠ( x, ∞ ) = γ ν Γ( ν ) x ν − e − γx + ϕ γ ν Γ( ν ) Z ∞ x y ν − e − γy dy, where the constants ϕ, γ > ν ∈ (0 , L − isthe sum of two independent subordinators, one of which is a compound Poissonprocess with gamma distributed jumps, the other has infinite activity and is ofthe so called tempered-stable type. Clearly Π also describes the L´evy measureof − Λ too.According to [9], the process Λ belongs to the Gaussian Tempered StableConvolution class and moreover, ψ Λ ( θ ) = ( θ − ϕ ) (cid:18) − (cid:18) γγ + θ (cid:19) ν (cid:19) for θ ≥ 0. In particular δ Λ = 1 and Φ Λ (0) = ϕ . It is a straightforward exericseto show that E (Λ ) = ψ ′ Λ (0+) = − ϕ νγ and this implies that µ = 11 + ϕν/γ < , as required.From [9] we also know that W ( x ) = e ϕx + γ ν e ϕx Z x e − ( γ + ϕ ) y y ν − E ν,ν ( γ ν y ν ) dy where E α,β ( x ) := X n ≥ z n Γ( αn + β )is the two parameter Mittag-Leffler function.We may now deduce from the theory presented earlier that, for example, P d ( Q ∗ ≤ x ) = 1 − e − ϕx + γ ν R x e − ( γ + ϕ ) y y ν − E ν,ν ( γ ν y ν ) dy γ ν R x e − ( γ + ϕ ) y y ν − E ν,ν ( γ ν y ν ) dy and P d ( Q ∗ > x ) ∼ (cid:18) − (cid:18) γγ + ϕ (cid:19) ν (cid:19) e − ϕx . Acknowledgement. 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