Palm theory of random time changes

Palm distributions are basic tools when studying point processes, queueing systems, fluid queues or random measures. The framework usually varies with the random phenomenon of interest, but commonly a one-dimensional group of ineasure-preserving shifts is present. In this research we study random time changes (RTC's, right-continuous and non-decreasing functionals which pass the zero-level at 0), appropriate to characterize all above systems. We assume the existence of a two-dimensional family of shifts which behaves consistently like a group along the extended graph of the RTC. In canonical settings this assumption is trivially satisfied. From this family we derive two one-dimensional groups of shifts and assume for one of them that the elements preserve the underlying distribution P. As a consequence, the elements of the second group preserve the detailed Palm distribution. This DPD has a very natural interpretation and satisfies a duality criterion: the DPD of the DPD gives P in return. For this framework of time changes, we also consider the version of the n classical" Palm distribution. The relationship with the DPD is studied. We prove that Palm theory for random measures is indeed included in Palm theory for RTC's. In a theoretical application we consider the important non-simple (marked) point process case and define another distribution ofPalm type. It can discriminate between (the marks of) simultaneous occurrences and has nice stationarity properties.


Introduction
Palm theory is especially known because of its merits for the study of stationary queueing systems. See, e.g., Franken et al. (1982), Brandt et al. (1990), Baccelli 8c Brémaud (1994), and Sigman (1995). In the presence of a group of ineasure-preserving, onedimensional time-shifts the theory considers the relationship between two distributions, a time-stationary distribution and a Palm distribution (PD). Both describe the stochastic behavior of the system. The first as it is seen from a randomly chosen time and the second from a randomly chosen arrival (point) or arrival epoch. In classical queueing theory there are two basic frameworks to describe the process of arriving customers, each with its own merits. In Franken et al. (1982) the realizations are characterized as integervalued measures counting the numbers of arrivals. For simple queueing systems (i.e., only single arrivals) a natural group of point-shifts exists which is stationary under the PD. An advantage of this framework is that the setting of counting measures naturally generalizes to a setting of ineasures to cover the more general framework needed for Palm theory for modern fluid queues and random measures. See, e.g., Schmidt 8L Serfozo (1994) and Miyazawa (1994). However, a disadvantage is that this framework and the PD are less appropriate in the case of non-simple queueing systems. The PD looses its nice stationarity properties and it cannot discriminate between two simultaneous arrivals within a batch of arrivals. (Cf. Section 1.3.7 in Franken et al. (1982) and page 87 in Kdnig 8L Schmidt (1992).) In Brandt et al. (1990) the realizations of a system of arriving customers are characterized as sequences of non-decreasing times which correspond to arrival epochs of customers. The distribution P defined on page 82 of this reference is of Palm type and can discriminate between (the marks of) two simultaneous arrivals.
However, this framework can only be used to describe point process systems and cannot be generalized to a more general framework for fluid queues.
The general framework of ineasures mentioned above can equivalently be characterized by the set of right-continuous and non-decreasing functions g on R with g(0) -0. The PD of a random functional with realizations in this set follows immediately from (1.4) of Geman 8z Horowitz (1973) and is equivalent to the PD of the corresponding random measure. So, this framework of functions has the same advantages and disadvantages as the framework of ineasures. As in this last reference, we point out that in general the relationship between the time-stationary distribution and its PD is not dual.
In the present research we will consider a frame-work and a distribution of Palm type which overcome all disadvantages mentioned above. This DPD, detailed Palm distribution, restores duality and stationarity. In a slightly modified form it was first mentioned in Miyazawa (1994), on a smaller v-field from a more applied point of view.
We will study the properties of this (general) DPD. It behaves like "the mother of all PD's" in the sense that all well-known distributions of Palm type follow immediately from it, in a natural and very intuitive way. To compare these distibutions of Palm type, we will write them all as integrals along the vertical axis. This turns out to be more natural. Essentially, we generalize Geman 8L Horowitz' set of realizations to a set G by replacing g(0) -0 for the right-continuous and non-decreasing functions g by g(0-) C 0 G g(0). We will consider random time changes (RTC's) A, random functionals on (S2,.P, P) with values in G, and study Palm theory for such random phenomena. In contrast with the frameworks mentioned above, a two-dimensional family O of transformations (shifts) on S2 will be considered. It will be assumed that this family of shifts behaves like the natural two-dimensional family in the canonical case, the case that A is the identity on S2 -G. That is, O behaves like a group along the extended graph (i.e., the graph extended with the vertical jump-parts) of A(., w), in a consistent way. Assuming time-stationarity (as usual, one-dimensional along the horizontal axis) with respect to P, we define the detailed Palm distribution (DPD) Põ f P with respect to A. With A'(.,w) the right-continuous inverse of A(.,w), it is proved that the pairs (A, P) and (A', P~) are dual: taking the DPD of the DPD gives P in return. Both P and P~describe the stochastic behavior of the system. The first as it is seen on the graph of A from a position with the first coordinate chosen at random, the second as it is seen on the extended graph of A from a position with the second coordinate chosen at random. Stating it otherwise, P~describes the stochastic behavior of the system as it is seen on the graph of A' from a position with the first coordinate chosen at random. From this heuristic description of the relationship between P and its DPD it immediately becomes clear how -for the present framework-the "ordinary" PD's of special systems should be defined. For instance in the case of a non-simple queueing system, a PD which can discriminate between two simultaneous arrivals and which has nice stationarity properties follows immediately from the DPD.
In Section 2 we first introduce the framework and give the definition of a random time change A. It is assumed that a two-dimensional family O of transformations exists which behaves as a group along the extended graph of A, in a consistent way. In a canonical setting this assumption is naturally satisfied. The family O induces one dimensional groups 9 and~of transformations (shifts) on f2, the first on the horizontal axis and tlie second on the vertical axis. The relationship between B, rl, A, and its generalized, right-continuous inverse A' is considered in a few lemma's. In Section 3 we formulate the further assumptions, that B is stationary with respect to P and that the (possibly random) long-run average limi-.~A(t)~t is positive and finite. The detailed Palm dis-tribution P~is defined and it is proved that the group~is stationary w.r.t. it. The intuitive interpretation of P~mentioned before is formalized. While A(0) is zero a.s. under P, it turns out that under P~it is -given the jumpsize A(0) -A(0-) at zero-uniform [0, A(0) -A(0-)] distributed. We also consider the Palm distribution Po which in the present framework is most similar to the PD's of the textbooks and papers mentioned above. Just like P~, this PD is also defined in terms of an integral along the vertical axis. In the canonical case it simply arises from the DPD by shifting the origin (on This duality principle can be used to derive results for P~from similar results for P(and více versa). The duality between P and P~, and the simple relationship between the two Palm distributions P~and Po is used to obtain a general inversion formula to express P in terms of Po.
In Section 5 we show that Palm theory for random measures is indeed included in Palm theory for RTC's. From a random measure satisfying the usual assumptions (see, e.g., Schmidt óe Serfozo (1994) and Miyazawa (1994)), we constructively create an RTC which satisfies the assumptions of the present research, without additional assumptions. The (ordinary) PD of this RTC corresponds to the PD of the random measure we started with. Section 6 can be seen as a theoretical application. It shows that the present framework is appropriate in cases where the counting measure framework is less suitable.  Let (S2,.F) be a measurable space. A random time change (shortly, RTC) A is a measurable mapping S2 -~G. For w E S2 we will write A(., w) for the corresponding function in G and A(t,w) for its value in t E R. The generalized inverse of A(.,w) is denoted by A'(.,w). So, A' is another random time change. The extended graphs of A(.,w) and A'(.,w) are shortly denoted by I'(w) and I"(w), respectively. In this context we will usually use s and t to denote elements of the horizontal axis of I'(w), and x and y for elements of the vertical axis.
Let (52,.~) be a measurable space such that f2~52 and~fl 52 -.~. We call (52,.~) an extension of (52,.~). Let 0-{O1t,I~:(t,x) E R2} be a family of transformations on S2, not necessarily a group. I.e., O~t,xl(w) The assumption below expresses that the (random) extended graph I' of A is consistent with O, and that the family O behaves itself on I' as a group. Assume: (i) For all w E S2, ( t, x) E I'(w) and (s, y) E I'(O~t,z~w) we have: Assumption (i) is motivated by canonical settings (useful in applications) as in the following example.  we have:  In view of this lemma, we denote Z~BI and Z~nl by a single notation Z. Note that, as an immediate consequence of the lemma, for all Z-measurable functions f : fl -~R and all t, x E R.
In the next sections we will occasionally use the left-continuous inverse g-1 of g E G, defined by g-1(x) -inf{s E R: g(s)~x},~E R. Let g" be the measure generated 6y g. I.e.,

g'((s, t]) :-g(t) -g(s), s c t.
(2. 2) The following lemma enables us to transform integrals with respect to g', on the horizontal axis, into Lebesgue-integrals on the vertical axis. It will be proved in the Appendix.
Two distributions of Palm type An RTC is a right-continuous and non-decreasing random functional, not necessarily zero at 0 but passing the zero-level at 0. For such a functional we define two Palm distributions (PD's). The detailed PD has a nice stationarity property, the "ordinary" PD is very similar to the well-known PD for random measures. Heuristically, both describe the stochastic behavior of the functional as it is seen from a randomly chosen position on the extended graph; however, the randomness procedures differ. The relationship between the two distributions is studied. Things are generalized by considering marked time changes: RTC's accompanied by a stochastic process on their extended graphs.
We first introduce a probability measure P on (S2,.P). Apart from Assumption (i), we will also assume that the family B is stationary with respect to P, i.e.:  We will write Eo for expectations under Po. In the next theorem the relationship between PD and DPD is studied. It is proved in the Appendix.

S((A'(x), x)); t, x E R. It is an easy exercise to prove that the stochastic processes
Sl and S2 satisfy

Á' n~(o)
Theorem 4.1. The detailed Palm distribution of Pn with respect to A' is equal to P.
Especially, for A E .P,  Furthermore, we assume that (iii-a) Q(0 G Ao G oo) -1.

Proof. Since ( i)'-(iii)' are satisfied, we can apply Theorem 3.2 replacing A by A', P by
Here Ao is the long-run average E(Ao(1)~Zo) -IimAo(t)~t with Zo the invariant v-field :-.õ f r. Similar to Schmidt 8z Serfozo (1994), we define the Palm distribution Qo of Q with respect to Aó by As in (3.7), we use the random intensity; see Sigman (1995) and Nieuwenhuis (1997). Let w-(wo, z) be an element of S~. For s, t, x E R we define:

A'((s,t],w) :-A(t,w) -A(s,w) for s C t.
Next, we identify f2o and f2o x{0}. With this identification, A and A' are extensions of Ao and Aó. Note, however, that f2 -12o if Ao(.,w) is continuous on R for all w E ,-i (6.12)

A'(P,w) -Aó(B,wo) for all B E Ci(R), and that the random function
As n-~oo, the LHS tends to P~(A~Z), both P~-a.s. and P-a.s. The RHS of (6.12) tends i to É~~f lA o~zdx~Z~~, both P~-a.s. and P-a.s. Since P-P~on Z and P-P~oǹ o n Z~, we obtain both sides of (a) as limits of -f P(~~lA)dy as n-~oo. So, the two sides So, the sequeiice (m~)~E~is stationary under P~. In view of (6.6) this result is intuitively clear (and can also be proved from it), at least in the canonical setting. Under P~we can, for instance, consider the probability that the mark of the first customer in order has some property, even in the case that customers arive in batches.
Let w E S2 be such that B,w E A. Note that Bow -9-,(B,w) belongs to A because of the above arguments. By (A.2), with x--A(O,w), we obtain that rtow -gr(9ow) belongs to A. Again by (A.2), with w' -w and x-0, we conclude that w E A. O Proof of Lemma 2.4. It is an easy exercise to prove that, for all s, t E R with s G t,