1 Introduction
This paper focuses on the Lindley type stochastic recursionwhere \(\left[ x\right] ^+=\max \{x,0\}\) for \(x\in {\mathbb R}\), and \(\{V_i\}_{i\in {\mathbb N}_0}\) and \(\{Y_i\}_{i\in {\mathbb N}_0}\) are independent sequences of i.i.d. (independent, identically distributed) random variables. The analysis of stochastic recursions has received much attention in the applied probability literature. This holds in particular for stochastic recursions of the autoregressive type, owing to their wide applicability across various scientific domains including biology, finance, and engineering [7, 9, 14].
$$\begin{aligned} W_{i+1} = [V_iW_i+Y_i]^+, \quad i=0,1,\ldots , \end{aligned}$$
(1)
An important subclass of first-order autoregressive models corresponds to the case in which the \(\{V_i\}_{i\in {\mathbb N}_0}\) are constant, i.e., a stochastic process defined through the recursionfor a sequence of i.i.d. random variables \(\{Y_i\}_{i\in {\mathbb N}_0}\) and a scalar a, with \(W_0\) being given. When the quantities \(W_i\) cannot attain negative values, it becomes natural to study the truncated counterpart of (2), i.e., the recursionWhen \(a=1\) we recover the classical Lindley recursion describing the waiting time in the G/G/1 queue, with \(Y_i\) representing the difference between the i-th service time and the (\(i+1\))-st interarrival time. The case of \(a \in (0,1)\) was studied in detail in [7], whereas the case \(a=-1\) is covered by [23]. It should be observed that, while from the analysis it is clear that a is assumed to be positive in [7], the introduction of that paper incorrectly states that \(|a| < 1\).
$$\begin{aligned} W_{i+1} = aW_i +Y_i, \quad i=0,1,\ldots , \end{aligned}$$
(2)
$$\begin{aligned} W_{i+1} = [aW_i +Y_i]^+, \quad i=0,1,\ldots . \end{aligned}$$
(3)
Advertisement
By studying (1), we significantly extend the analysis of the Lindley recursion as well as the analysis of the stochastic recursion (3). Our results focus on three different choices of \(\{V_i\}\). In Model I, the \(V_i\) are either negative or equal to the positive constant a. Here, we demand that both the positive and the negative parts of the \(Y_i\) have a rational Laplace–Stieltjes transform (in the sequel abbreviated to LST). In Model II, the \(V_i\) are negative random variables. A detailed analysis is shown to be possible as long as the positive part of the \(Y_i\) has a rational LST. While Model I to a large extent contains Model II, we prefer to give a separate analysis of both models, to make the reader familiar with the specific mathematical intricacies due to \(V_i\) being negative (Model II) and \(V_i\) being a positive constant (the model in [7]). Moreover, the extra assumption that the negative part of the \(Y_i\) has a rational LST plays an important part in the analysis of Model I, precluding the possibility to simply obtain the results for Model II from those for Model I. Finally, in Model III, the \(V_i\) are uniformly distributed on [0, 1], and the negative part of the \(Y_i\) is exponentially distributed; this case requires an entirely different approach.
The rationality assumptions are natural in the light of the existing theory that has been developed for the G/G/1 queue. While in principle the waiting-time distribution in the general G/G/1 queue can be obtained via a Wiener–Hopf decomposition (cf. [11, Chapter II.5]), the solution is a rather implicit one, unless one makes rationality assumptions on either the interarrival or the service-time LST. In addition, it can be argued that the distribution of any nonnegative random variable can be approximated arbitrarily closely by the distribution of a random variable with a rational LST [1, Ch. III], so that a restriction to random variables with rational LST leads to just a minor loss of generality.
Notable studies of stochastic recursions are [4, 13, 16]; see in addition [3]. For the non-reflected case, stochastic recursions of the form \(W_{i+1} = V_iW_i + Y_i\) have been studied frequently, partly under the name ‘Vervaat perpetuity’; we mention [9, 12, 14, 17, 19, 22]. For the reflected case, [8] considers another generalization of the Lindley recursion, by replacing \(V_iW_i\) in (1) by \(S(W_i)\), where \(\{S(t)\}_{t\geqslant 0}\) is a Lévy subordinator. A model that is similar to the present model has been discussed in [24]. It is noted, though, that [24] primarily focuses on stability questions, limit theorems and questions related to queuing applications, whereas our primary focus lies on the derivation of results for the transient and stationary distribution of the process under investigation. Also related is the model in [6]; there (1) is considered with \({{\mathbb P}}(V_i=1)=p\), \({{\mathbb P}}(V_i=-1)=1-p\).
The main contributions of the present paper are the following: For Models I and II, we state and solve a Wiener–Hopf boundary value problem, which allows us to study the transient behavior of the \(\{W_i\}_{i\in {\mathbb N}_0}\) process. In particular, we obtain an expression for the objectwhich can be interpreted as the generating function of the LST of the \(W_i\), but also (up to the multiplicative constant \(1-r\)) as the LST after a geometrically distributed time. The transient behavior of the \(\{W_i\}_{i\in {\mathbb N}_0}\) process is also obtained for Model III, using a different argument. The stability condition of each of the three models is discussed, and the steady-state distribution of the \(\{W_i\}_{i\in {\mathbb N}_0}\) process is also determined.
$$\begin{aligned} \sum _{i=0}^{\infty } r^i \,{{\mathbb E}}(\mathrm{e}^{-sW_i}), \end{aligned}$$
Advertisement
The remainder of the paper is organized as follows: Section 2 presents the description of the three models and some preliminaries. Sections 3, 4 and 5 are devoted to the transient and steady-state analysis of, respectively, Models I, II, and III. Section 6 contains a discussion and provides suggestions for further research.
2 Model description and preliminaries
The main object of study is the stochastic recursionwhere \(\{V_i\}_{i\in {\mathbb N}_0}\) and \(\{Y_i\}_{i\in {\mathbb N}_0}\) are sequences of i.i.d. random variables, which are in addition independent of each other. The initial state of the process is assumed to be \(W_0=w\in {\mathbb R}^+\). We write \(V\) and Y for generic random variables distributed as \(V_0\) and \(Y_0\) respectively.
$$\begin{aligned} W_{i+1} = \left[ V_iW_i + Y_i\right] ^+, \quad i=0,1,\ldots , \end{aligned}$$
(4)
In this paper, we discuss the following three variants of the model:In each of the cases, we will assume that the \(Y_i\) are decomposed as \(B_i-A_i\), with sequences \(\{A_i\}_{i\in {\mathbb N}_0}\) and \(\{B_i\}_{i\in {\mathbb N}_0}\) of i.i.d. nonnegative random variables. In addition, depending on the chosen model, the random variables \(A_i\) and/or \(B_i\) are assumed to have a rational LST.
$$\begin{aligned} \begin{array}{ll} {\textit{Model\,I}}:&{} {{\mathbb P}}(V=a)=p,\ {{\mathbb P}}(V<0)=1-p,\quad a>0,\ p\in (0,1);\\ {\textit{Model\,II}}: &{} {{\mathbb P}}(V<0)=1;\\ {\textit{Model\,III}}:&{} {{\mathbb P}}(V<x) = x, \quad 0 \leqslant x \leqslant 1. \end{array} \end{aligned}$$
We start with investigating the stationary behavior of \(\{W_i\}_{i\in {\mathbb N}_0}\). We always assume that both \({{\mathbb E}}|V|\) and \({{\mathbb E}}|Y|\) are finite. The first result was given in [24] and covers most cases of interest.
Theorem 1
[24] If one of the following conditions holds, then \(W_i\) tends weakly to a proper limit W as \(i\rightarrow \infty \): Moreover, \(W_i\) converges weakly to a possibly improper limit W as \(i\rightarrow \infty \) if \(V>0\) a.s., \({{\mathbb E}}(\log |V|)=0\), and \(W_0=0\). If additionally \(V=1\) a.s. then W is proper for \({{\mathbb E}}(Y)<0\) and improper for \({{\mathbb E}}(Y)>0\).
(C1)
\({{\mathbb P}}(V<0)>0\) and \({{\mathbb P}}(Y\leqslant 0)>0\),
(C2)
\(V\geqslant 0\) a.s. and \({{\mathbb P}}(V=0)>0\),
(C3)
\(V>0\) a.s. and \({{\mathbb E}}(\log |V|)<0\).
It follows straightforwardly from the regenerative structure of \(W_i\), and the proof of the above theorem in [24], that in cases (C1) and (C2) the limit W is unique. Obviously, under the conditions of the theorem, the limiting random variable W fulfils the associated distributional identity \(W{=}_{\mathrm{d}}\left[ VW + Y\right] ^+\). Regarding the above condition (C3), we add the following observation.
Theorem 2
In order to have convergence of \(W_i\) to a proper unique limit W as \(i\rightarrow \infty \), it is sufficient to have \({{\mathbb E}}(\log |V|)<0\), which in turn is implied by \({{\mathbb E}}|V|<1\).
Proof
Recursion (4) can be written as a random iteration \(W_{i+1}=f_{\theta _i}(W_i)\) with \(f_{\theta _i}(x)=\left[ xV_i+Y_i\right] ^+\), \(\theta _i=(V_i,Y_i)\). This means that \(f_\theta (\cdot )\) enjoys the Lipschitz propertywith random Lipschitz constant \(K_\theta =|V|\). As a result, [13, Thm. 1.1] is applicable. By Jensen’s inequality, \({{\mathbb E}}\log |V|\leqslant \log {{\mathbb E}}|V|\) and so \({{\mathbb E}}|V|<1\) implies the condition \({{\mathbb E}}(\log |V|)<0\). \(\square \)
$$\begin{aligned} |f_{\theta }(x)-f_{\theta }(y)|\leqslant K_\theta |x-y|, \end{aligned}$$
The case where \({{\mathbb P}}(V<0)>0\) and \(Y\geqslant 0\) a.s., which was omitted in [24], is more involved due to the fact that the process might not be aperiodic, even if Y is not deterministic. As an example suppose that the distribution of Y is supported on [1, 2] and that \(V\leqslant -2\) a.s. If \(W_0=0\), then \(W_1\in [1,2]\), \(W_2=0\), \(W_3\in [1,2]\), entailing that the process alternates between the set \(\{0\}\) and a value in [1, 2]. On the other hand, if \(W_0>2\), then \(W_1=0\), \(W_2\in [1,2]\) and so on. As a consequence, there is no convergence \(W_i\Rightarrow W\) as \(i\rightarrow \infty \). However, regarding the existence of a stationary distribution, we can show the following.
Theorem 3
If \({{\mathbb P}}(V\leqslant 0)>0\) and \(Y\geqslant 0\) a.s., then there is convergence of \(W_i\) to a stationary random variable W as \(i\rightarrow \infty \).
Proof
We define a majorizing process by \(M_0:=W_0\) and \(M_{i+1}:=\left[ V_i\right] ^+M_i+Y_i\). Then, \(W_i\leqslant M_i\), \(i=0,1,2,\ldots \) and for \(\{M_i\}_{i\in {\mathbb N}_0}\) we have \({{\mathbb E}}(\log |\left[ V\right] ^+|)=-\infty <0\) since \({{\mathbb P}}(\left[ V\right] ^+=0)>0\). So by Theorem 2 it follows that \(M_i\Rightarrow M\) as \(i\rightarrow \infty \) for some limiting random variable \(M\). Hence, \(\{M_i\}_{i\in {\mathbb N}_0}\) is tight, and then, the sequence \(\{W_i\}_{i\in {\mathbb N}_0}\) is tight as well. Thus, [15, Thm. 4] guarantees the existence of a stationary distribution, for \(\{W_i\}_{i\in {\mathbb N}_0}\). \(\square \)
We end this section with a lemma that forms the starting-point of the analysis of all three models. For this, we need to introduce some additional notation. For a given nonnegative random variable X, we write \(\Phi _{X}(s)={{\mathbb E}}e^{-sX}\) for its LST, defined at least for \(\mathrm{Re}\,s\geqslant 0\). We say that \(\Phi _{X}\in {\mathbb Q}[s_1,s_2,\dots ,s_n]\) if X has a rational LST with poles at \(s_1,s_2,\dots ,s_n\), i.e., if \(\Phi _{X}(s)\) is of the formwhere \(D_{X}(s)=\prod _{i=1}^n (s-s_i)\) and \(N_{X}(s)\) is a polynomial of degree at most \(n-1\) not sharing zeros with \(D_{X}(s)\). Note that this implies that \({{\mathbb P}}(X=0)=\lim _{s\rightarrow \infty }\Phi _{X}(s)=0\). With this notation we then have, for example, \(\Phi _{Y}(s)=\Phi _{B}(s)\,\Phi _{A}(-s)\). We also write \(D_{Y}(s)=D_{B}(s)\,D_{A}(-s)\) and \(N_{Y}(s)=N_{B}(s)\,N_{A}(-s)\) if A and B have rational LSTs of the form (5).
$$\begin{aligned} \Phi _{X}(s)=\frac{N_{X}(s)}{D_{X}(s)}, \end{aligned}$$
(5)
For a sequence \(\{X_i\}_{i\in {\mathbb N}_0}\) of random variables, we introduce the generating function, for \(r\in (0,1)\):Note that since \(\Phi _{VX}(s)=\int \Phi _{X}(ys)\,{{\mathbb P}}(V\in \mathrm{d}y)\), we haveThe following lemma plays a key role in our analysis. Define \(W^*_i:=\left[ V_iW_i+Y_i\right] ^-\), where \(\left[ x\right] ^-:=\min \{x,0\}\).
$$\begin{aligned} U_{X}(r,s)=\sum _{i=0}^\infty r^i\Phi _{X_i}(s). \end{aligned}$$
$$\begin{aligned} U_{VX}(r,s)=\int U_{X}(r,ys)\,{{\mathbb P}}(V\in \,\mathrm{d}y). \end{aligned}$$
(6)
Lemma 4
\(U_{W}(r,s)\) and \(U_{W^*}(r,s)\) are, for \(r\in (0,1)\) and \(\mathrm{Re}\,s = 0\), related via
$$\begin{aligned} U_{W}(r,s) = \mathrm{e}^{-sw}+ r\left( \Phi _{Y}(s) U_{VW}(r,s) + \frac{1}{1-r} - U_{W^*}(r,s)\right) . \end{aligned}$$
(7)
Proof
First observe that the basic identity \(\mathrm{exp}\left( \left[ x\right] ^+\right) = \mathrm{exp}\left( x\right) +1-\mathrm{exp}\left( \left[ x\right] ^-\right) \) applies. It thus follows from (4) thatMultiplying both sides of (8) by \(r^{i+1}\) and summing yields the identity (7). \(\square \)
$$\begin{aligned} \Phi _{W_{i+1}}(s) =\Phi _{V_iW_i+Y_i}(s) +1-\Phi _{W^*_i}(s), \quad i=0,1,\ldots . \end{aligned}$$
(8)
3 Model I: the mixed case
In this section, we consider the following model: We start from the recursion (4), and we assume that \(V=a\) for \(a>0\) with probability p and \(V<0\) with probability \(1-p\). LetWe assume that both A and B have a rational LST. Summarizing, we impose the conditions
$$\begin{aligned} V^-=(V\,|\,V<0). \end{aligned}$$
(A)
\({{\mathbb P}}(V=a)=p,\ {{\mathbb P}}(V<0)=1-p,\ a>0,\ p\in (0,1)\),
(B)
\(\Phi _{B}\in {\mathbb Q}[s_1,\dots ,s_\ell ]\) with \(\mathrm{Re}\,s_j<0\) for \(j=1,\dots ,\ell \),
(C)
\(\Phi _{A}\in {\mathbb Q}[t_1,\dots ,t_m]\) with \(\mathrm{Re}\,t_i<0\) for \(i=1,\dots ,m\).
Theorem 5
Suppose that the conditions (A), (B), and (C) hold. Then, for \(r\in (0,1)\),
1.
if \(a=1\) then where while the remaining constants \(a_1(r),\dots ,a_{m+\ell }(r)\) can be determined from the linear systems (16) and (18) that will be given below.
$$\begin{aligned} U_{W}(r,s) =\frac{D_{Y}(s)\,\mathrm{e}^{-sw}+\sum _{k=0}^{m+\ell } a_k(r) s^k}{D_{Y}(s)-rp\,N_{Y}(s)}; \end{aligned}$$
(9)
$$\begin{aligned} a_0(r) = \frac{r}{1-r} (1-p) (-1)^{\ell +m}\prod _{j=1}^\ell s_j \prod _{i=1}^m t_i, \end{aligned}$$
(10)
2.
if \(a\not =1\) then where \( a_0(r)\) is as in (10) and the remaining constants \(a_1(r),\dots ,a_{m+\ell }(r)\) can be determined from the linear systems (26) and (27) that will be given below.
$$\begin{aligned} U_{W}(r,s) = \sum _{h=0}^{\infty } \left( \mathrm{e}^{-a^hsw} + \frac{\sum _{k=0}^{m+\ell } a_k(r) (a^hs)^k}{D_{Y}(a^hs)}\right) (rp)^h \prod _{j=0}^{h-1}\Phi _{Y}(a^js), \end{aligned}$$
(11)
Proof
In this situationThen (7) becomes, after multiplication by \(D_{Y}(s)\),Now the following are true: Both sides are well defined at the boundary \(\mathrm{Re}\,s=0\). Determination of the unknown functions \(U_{W}(r,s)\) and \(U_{W^*}(r,s)\) from (13) and conditions (i), (ii), and (iii) is a Wiener–Hopf boundary value problem of a type that has been extensively studied in queueing theory before, cf. the expository paper [10]. By introducing a function G(r, s) that is equal to the left-hand side of (13) for \(\mathrm{Re}\,s \geqslant 0\) and to the right-hand side of (13) for \(\mathrm{Re}\,s \leqslant 0\), we have a function that is analytic in the whole s-plane, and that for large s is \(O(s^{m+ \ell })\). Liouville’s theorem [21, p. 85] now states that both sides of (13), in their respective half-planes, are equal to the same \((m+ \ell )\)-th degree polynomial in s. In other words, for \(\mathrm{Re} \, s \ge 0\),and, for \(\mathrm{Re}\,s \leqslant 0\),Taking \(s=0\) in either (14) or (15) yields, after a straightforward calculation, the expression for \(a_0(r)\) in (10). Next we set \(s=s_j\), \(j=1,\dots ,\ell \), in (15). Since \(D_{B}(s_j)=0\) it follows thatWe thus have obtained \(\ell \) linear equations in the remaining \(m+\ell \) unknown \(a_k(r)\); however, they are expressed in the yet unknown function \(U_{W}(r,\cdot )\).
$$\begin{aligned} U_{VW}(r,s) = p\, U_{W}(r,as) + (1-p) \int _{-\infty }^0 U_{W}(r,sy)\;{{\mathbb P}}(V^-\in \mathrm{d}y) . \end{aligned}$$
(12)
$$\begin{aligned}&D_{Y}(s) \big (U_{W}(r,s) - \mathrm{e}^{-sw}\big ) - r pN_{Y}(s)\, U_{W}(r,as)\nonumber \\&\quad = r(1-p)\,N_{Y}(s) \int _{-\infty }^0 U_{W}(r,sy)\;{{\mathbb P}}(V^-\in \mathrm{d}y)+ rD_{Y}(s)\left( \frac{1}{1-r} - U_{W^*}(r,s)\right) . \end{aligned}$$
(13)
(i)
the left-hand side of (13) is analytic in \(\mathrm{Re}\,s >0\) and continuous in \(\mathrm{Re}\,s \geqslant 0\),
(ii)
the right-hand side of (13) is analytic in \(\mathrm{Re}\,s <0\) and continuous in \(\mathrm{Re}\,s \leqslant 0\),
(iii)
for large s, both sides are \(O(s^{m+\ell })\) in their respective half-planes.
$$\begin{aligned} D_{Y}(s) \big (U_{W}(r,s) - \mathrm{e}^{-sw}\big ) - r pN_{Y}(s) U_{W}(r,as) =\sum _{k=0}^{m+\ell } a_k(r) s^k, \end{aligned}$$
(14)
$$\begin{aligned}&r(1-p)\,N_{Y}(s) \int _{-\infty }^0 U_{W}(r,sy)\;{{\mathbb P}}(V^-\in \mathrm{d}y) + rD_{Y}(s)\left( \frac{1}{1-r} - U_{W^*}(r,s)\right) \nonumber \\&\quad =\sum _{k=0}^{m+\ell } a_k(r) s^k. \end{aligned}$$
(15)
$$\begin{aligned} r(1-p)N_{Y}(s_j) \int _{-\infty }^0 U_{W}(r,s_jy)\;{{\mathbb P}}(V^-\in \mathrm{d}y) = \sum _{k=0}^{m+\ell } a_k(r) s_j^k , \quad j=1,\dots ,\ell . \end{aligned}$$
(16)
We turn to (14), which provides a relation between \(U_{W}(r,s)\) and \(U_{W}(r,as)\). As it turns out, we have to distinguish between the two cases \(a=1\) and \(a\not =1\):
This finishes the proof. \(\square \)
\(\circ \)
Case i: For \(a=1\), after division by the denominators, Relation (14) can be rewritten as Cohen [11], in his study of the \(\hbox {K}_m\)/G/1 queue, proves that the term between brackets in the left-hand side of (17) has m zeroes \(\delta _1(r),\dots ,\delta _m(r)\) in the right half plane \(\mathrm{Re}\,s > 0\). The analyticity of \(U_{W}(r,s)\) for \(\mathrm{Re}\,s \geqslant 0\) now implies that the right-hand side of (17) must be zero for all these m zeroes. This results in the m linear equations Formula (16) contains \(\ell \) more equations in the \(a_k(r)\). Relying on (17), we can rewrite it into for \(j=1,\dots ,\ell \). From this, we obtain where, for \(j=1,\dots ,\ell \),
$$\begin{aligned} U_{W}(r,s)\big (1-rp\Phi _{Y}(s)\big ) =\mathrm{e}^{-sw}+\frac{\sum _{k=0}^{m+\ell } a_k(r) s^k}{D_{Y}(s)}. \end{aligned}$$
(17)
$$\begin{aligned} \sum _{k=0}^{m+\ell } \delta _i^k(r)a_k(r) = -\mathrm{e}^{-\delta _i(r) w}D_{Y}(\delta _i(r)), \quad i=1,\dots ,m. \end{aligned}$$
(18)
$$\begin{aligned} r(1-p)N_{Y}(s_j) \int _{-\infty }^0 \frac{\mathrm{e}^{-s_jyw}+\sum _{k=0}^{m+\ell } a_k(r) \frac{(s_jy)^k}{D_{Y}(s_jy)}}{1-rp\Phi _{Y}(s_jy)}\;{{\mathbb P}}(V^-\in \mathrm{d}y) = \sum _{k=0}^{m+\ell } a_k(r) s_j^k, \end{aligned}$$
(19)
$$\begin{aligned} \sum _{k=0}^{m+\ell } c_{k}(r,s_j)\,a_k(r)=\int _{-\infty }^0 \frac{\mathrm{e}^{-s_jyw}}{1-rp\Phi _{Y}(s_jy)}\;{{\mathbb P}}(V^-\in \mathrm{d}y),\quad j=1,2,\dots ,\ell ,\nonumber \\ \end{aligned}$$
(20)
$$\begin{aligned} c_{k}(r,s_j)=\frac{s_j^k}{r(1-p)\,N_{Y}(s_j)}-\int _{-\infty }^0 \frac{(s_jy)^k}{D_{Y}(s_jy)-rp\,N_{Y}(s_jy)}\;{{\mathbb P}}(V^-\in \mathrm{d}y). \end{aligned}$$
\(\circ \)
Case ii: For \(a<1\), Relation (14) has the same structure as [7, Formula (2.3)]. Proceeding in a similar way as in [7], we write with For \(a\not =1\) iteration of (21) yields where convergence of the infinite sum can be proven using the d’Alembert test. Indeed, for \(a<1\) the limit as \(h\rightarrow \infty \) of the ratio of two successive terms is while for \(a>1\), \(K(r,a^hs)\rightarrow 0\) and \(|L(r,a^hs)|\rightarrow a_{m+\ell }(r)\), causing divergence of the left-hand side in (24) to infinity. Insertion of (22) in (23) gives (11). The only unknowns are \(a_1(r),\dots ,a_{m+\ell }(r)\). We obtain m linear equations in the unknown \(a_k(r)\) by observing that substitution of \(s =-t_i\), \(i=1,\dots ,m\), in (14) results in the following identity: Substituting the right-hand side of (11), with \(s = -at_i\), into (25) now gives, for \(a\not =1\), the m linear equations for \(i=1,\dots ,m\). The remaining \(\ell \) equations are provided by substituting (11) into (16), yielding for \(j=1,\dots ,\ell \), where
$$\begin{aligned} U_{W}(r,s) = K(r,s) \,U_{W}(r,as) + L(r,s), \end{aligned}$$
(21)
$$\begin{aligned} K(r,s) := rp\,\Phi _{Y}(s) , \quad L(r,s) := \mathrm{e}^{-sw} + \frac{\sum _{k=0}^{m+\ell } a_k(r) \,s^k}{D_{Y}(s)} . \end{aligned}$$
(22)
$$\begin{aligned} U_{W}(r,s) = \sum _{h=0}^{\infty } L(r,a^hs) \prod _{j=0}^{h-1} K(r,a^js) , \end{aligned}$$
(23)
$$\begin{aligned} \lim _{h\rightarrow \infty }\Big \vert \frac{L(r,a^hs) }{L(r,a^{h+1}s)K(r,a^hs)}\Big \vert =\frac{1}{rp}>1, \end{aligned}$$
(24)
$$\begin{aligned} -rpN_{Y}(-t_i) U_{W}(r,-at_i) = \sum _{k=0}^{m+\ell } a_k(r) (-t_i)^k, \quad i=1,\dots ,m. \end{aligned}$$
(25)
$$\begin{aligned}&\sum _{k=0}^{m+\ell } (-t_i)^k a_k(r) \left( 1+rp\,N_{Y}(-t_i) \sum _{h=0}^{\infty }\frac{a^{k(h+1)}(rp)^h \prod _{j=0}^{h-1}\Phi _{Y}(-a^{j+1}t_i) }{D_{Y}(-a^{h+1}t_i)} \right) \nonumber \\&\quad =-rpN_{Y}(-t_i) \sum _{h=0}^{\infty } \mathrm{e}^{a^{h+1}t_iw}(rp)^h \prod _{j=0}^{h-1}\Phi _{Y}(-a^{j+1}t_i) \end{aligned}$$
(26)
$$\begin{aligned}&\sum _{k=0}^{m+l} d_{k}(r,s_j)a_k(r) \nonumber \\&=r(1-p)N_{Y}(s_j) \sum _{h=0}^{\infty }(rp)^h \int _{-\infty }^0 \mathrm{e}^{-a^hs_jyw} \prod _{i=0}^{h-1}\Phi _{Y}(a^is_jy) \;{{\mathbb P}}(V^-\in \mathrm{d}y) \end{aligned}$$
(27)
$$\begin{aligned}&d_{k}(r,s_j)\nonumber \\&\quad =s_j^k\left( 1-r(1-p)N_{Y}(s_j)\sum _{h=0}^{\infty }(rp)^h \int _{-\infty }^0 \frac{ (a^hy)^k \prod _{i=0}^{h-1}\Phi _{Y}(a^is_jy)}{D_{Y}(a^hs_jy)} \;{{\mathbb P}}(V^-\in \mathrm{d}y)\right) . \end{aligned}$$
(28)
The steady-state LST of W exists if \({{\mathbb P}}(B\leqslant A)>0\), cf. Theorem 1. It can be obtained by applying an Abelian theorem. For example in case (ii), in which \(a<1\), one getswithwhere \(a_k := \mathrm{lim}_{r \uparrow 1} (1-r) a_k(r)\).
$$\begin{aligned} \Phi _W(s) = \mathrm{lim}_{r \uparrow 1} (1-r) U_{W}(r,s) = \sum _{h=0}^{\infty } L(a^hs) \prod _{j=0}^{h-1} K(a^js), \end{aligned}$$
(29)
$$\begin{aligned} K(s) := p\Phi _Y(s), \quad L(s) := \frac{\sum _{k=0}^{m+l} a_k s^k}{D_Y(s)}, \end{aligned}$$
(30)
4 Model II: the negative case
The model we analyze in this section assumes that each \(V_i\) attains only negative values and that \(Y_i\) is the difference \(B_i - A_i\) of two independent nonnegative random variables, where \(B_i\) has a rational LST. In other words, we impose the conditions We no longer need to impose condition (C) regarding the rationality of \(\Phi _A(\cdot )\).
(A*)
\(V<0\) a.s.,
(B)
\(\Phi _{B}\in {\mathbb Q}[s_1,\dots ,s_\ell ]\) with \(\mathrm{Re}\,s_j<0\) for \(j=1,\dots ,\ell \).
Theorem 6
Suppose that the conditions (A*) and (B) hold. Then, for \(r\in (0,1)\),whereand the remaining constants \(a_1(r),\dots ,a_\ell (r)\) can be determined from the linear system (39) that will be given below.
$$\begin{aligned} U_{W}(r,s) = \mathrm{e}^{-sw} + \frac{\sum _{k=0}^\ell a_k(r) s^k}{D_{B}(s)}, \quad \mathrm{Re}\,s\geqslant 0, \end{aligned}$$
(31)
$$\begin{aligned} a_0(r)= \frac{r}{1-r}(-1)^\ell \prod _{j=1}^\ell s_j, \end{aligned}$$
(32)
Proof
Multiplying both sides of (7) by the denominator \(D_{B}(s)\) givesNow observe the following: At the boundary \(\mathrm{Re}\,s=0\), both sides are well defined, so that we again have a Wiener–Hopf boundary value problem. As before, the G(r, s) that is equal to the left-hand side of (33) for \(\mathrm{Re}\,s \geqslant 0\) and to the right-hand side of (33) for \(\mathrm{Re}\,s \leqslant 0\) is analytic in the whole s-plane, and \(G(r,s)=O(s^{\ell })\) for large s. According to Liouville’s theorem both sides of (33), in their respective half-plane, are equal to the same \((\ell )\)-th degree polynomial in s, i.e., for \(\mathrm{Re}\,s \geqslant 0\),for \(\mathrm{Re}\,s\geqslant 0\) andfor \(\mathrm{Re}\,s \leqslant 0\). We still need to determine the \(\ell +1\) unknown functions \(a_0(r),\dots ,a_\ell (r)\). Taking \(s=0\) in either (34) or (35) gives the expression in (32) for \(a_0(r)\). Next we take \(s=s_j\), \(j=1,\dots ,\ell \). We do this in (35), observing that \(\mathrm{Re}\,s_j<0\). Using that \(D_{B}(s_j)=0\) we thus obtainApplying (6), this identity can be rewritten intoUsing (34), Eq. (37) becomes, for \(j=1,\dots ,\ell \),We can rewrite this equation as follows: for \(j=1,\dots ,\ell \),One can determine the remaining unknowns \(a_1(r),\dots ,a_\ell (r)\) from this set of \(\ell \) linear equations. Subsequently, from (34), (31) follows. \(\square \)
$$\begin{aligned}&D_{B}(s)\big (U_{W}(r,s) - \mathrm{e}^{-sw}\big ) \nonumber \\&\quad = rN_{B}(s)\,\Phi _{A}(-s) \,U_{VW}(r,s) + rD_{B}(s)\left( \frac{1}{1-r} - U_{W^*}(r,s)\right) . \end{aligned}$$
(33)
(i)
the left-hand side of (33) is analytic in \(\mathrm{Re}\,s >0\) and continuous in \(\mathrm{Re}\,s \geqslant 0\),
(ii)
the right-hand side of (33) is analytic in \(\mathrm{Re}\,s <0\) and continuous in \(\mathrm{Re}\,s \leqslant 0\),
(iii)
for large s, both sides are \(O(s^l)\) in their respective half-planes.
$$\begin{aligned} D_{B}(s)\big (U_{W}(r,s) - \mathrm{e}^{-sw}\big ) =\sum _{k=0}^\ell a_k(r) s^k \end{aligned}$$
(34)
$$\begin{aligned} rN_{B}(s)\,\Phi _{A}(-s)\, U_{VW}(r,s) + rD_{B}(s)\left( \frac{1}{1-r} - U_{W^*}(r,s)\right) =\sum _{k=0}^\ell a_k(r) s^k \end{aligned}$$
(35)
$$\begin{aligned} rN_{B}(s_j)\,\Phi _{A}(-s_j)\,U_{VW}(r,s_j) = \sum _{k=0}^\ell a_k(r) s_j^k , \quad j=1,\dots ,\ell . \end{aligned}$$
(36)
$$\begin{aligned} rN_{B}(s_j)\,\Phi _{A}(-s_j)\, \int _{-\infty }^0 U_{W}(r,s_jy) \,{{\mathbb P}}(V\in \,\mathrm{d}y) = \sum _{k=0}^\ell a_k(r) s_j^k , \quad j=1,\dots ,\ell . \end{aligned}$$
(37)
$$\begin{aligned} rN_{B}(s_j)\,\Phi _{A}(-s_j) \int _{-\infty }^0 \left( \mathrm{e}^{-s_j yw} + \frac{\sum _{k=0}^\ell a_k(r) (s_jy)^k}{D_{B}(s_jy)}\right) \,\;{{\mathbb P}}(V\in \mathrm{d}y) = \sum _{k=0}^\ell a_k(r) s_j^k. \end{aligned}$$
(38)
$$\begin{aligned}&\sum _{k=0}^\ell a_k(r) s_j^k\left( 1 - rN_{B}(s_j)\,\Phi _{A}(-s_j) \int _{-\infty }^0 \frac{y^k}{\prod _{m=1}^\ell (s_jy - s_m)}\;{{\mathbb P}}(V\in \mathrm{d}y) \right) \nonumber \\&\quad = r\,N_{B}(s_j)\,\Phi _{A}(-s_j)\,\Phi _{V}(s_jw). \end{aligned}$$
(39)
If \({{\mathbb P}}(B\leqslant A)>0\) holds then condition (C1) is fulfilled, so \(W_n\) weakly converges to a proper limit. We obtain the steady-state behavior via an Abelian theorem for power series:where \(a_k := \lim _{r \uparrow 1} (1-r) a_k(r)\), for \(k=0,\dots ,\ell \). Using (32) and (39), we readily obtain the linear systemfor \(a_j\), \(j=1,\dots ,\ell \), while \(a_0 =(-1)^\ell \prod _{i=1}^\ell s_i\).
$$\begin{aligned} \Phi _{W}(s) = \lim _{r \uparrow 1} (1-r) \,U_{W}(r,s) = \frac{\sum _{k=0}^\ell a_k s^k}{D_{B}(s)}, \quad \mathrm{Re}\,s \geqslant 0, \end{aligned}$$
(40)
$$\begin{aligned} \sum _{k=0}^\ell a_ks_j^k \left( 1 - N_{B}(s_j)\Phi _{A}(-s_j) \int _{-\infty }^0 \frac{y^k}{\prod _{m=1}^\ell (s_jy - s_m)}\;{{\mathbb P}}(V\in \mathrm{d}y) \right) = 0 \end{aligned}$$
(41)
The mean of W directly follows by differentiation of (40): \(\Phi _{W}'(0)=(a_1D_B(0)-a_0D_B'(0))/D_B(0)^2\), with \(D_B'(0)= 1\) if \(\ell =1\) and \(D_B'(0) = -\sum _{i=1}^\ell s_i\) if \(\ell =2,3,\ldots \); hence,
$$\begin{aligned} {{\mathbb E}}(W)={\left\{ \begin{array}{ll} {\displaystyle \frac{1-a_1}{a_0}},&{}\ell =1;\\ -{\displaystyle \frac{a_1+\sum _{i=1}^\ell s_i}{a_0}},&{}\ell =2,3,\ldots . \end{array}\right. } \end{aligned}$$
(42)
Remark 1
It follows from (40) that W is a mixture of an atom at zero (with probability \(a_\ell \)) and \(\ell \) exponential terms. This is not surprising: as \(V_i<0\), the only way for \(W_{i+1}\) to be positive is to have \(B_i > A_i - V_iW_i\). Now use the fact that \(B_i\) has a phase-type distribution with \(\ell \) exponential phases, in combination with the memoryless property of the exponential distribution.
Remark 2
When \(\ell =1\), one obtains, using that \(a_0=-s_1\),For general \(\ell \), we have not been able to verify formally that the set of \(\ell \) linear equations (41) in the \(\ell \) unknowns \(a_1,\dots ,a_{\ell }\) has a unique solution (as they involve the zeroes \(s_j\) and the distribution of V in an intricate way); similarly for the set of equations (39) for \(a_1(r),\dots ,a_{\ell }(r)\). However, since \(W_i\) has a unique limiting distribution with LST \(\Phi _{W}(s)\) as \(i\rightarrow \infty \), there is no reason to suspect that anomalies in this set of equations will occur.
$$\begin{aligned} a_1 = \frac{1 - N_{B}(s_1) \Phi _{A}(-s_1) \int _{-\infty }^0 \frac{1}{y-1} {{\mathbb P}}(V\in \mathrm{d}y) }{1 - N_{B}(s_1) \Phi _{A}(-s_1) \left( \int _{-\infty }^0 \frac{1}{y-1} {{\mathbb P}}(V\in \mathrm{d}y) +1\right) } . \end{aligned}$$
Example 1
Suppose that B has an exponential distribution with mean \(1/\mu \). Then \(\Phi _{B}(s)=\mu /(s+\mu )\), \(\ell =1\) and \(s_1=-\mu \). Suppose also that \(V=-a\) a.s. with \(a>0\). We then obtainMultiplying with \((1-r)\) and letting \(r\uparrow 1\) yields the coefficientsso that the LST of W is given bywhere the last equality follows from \({{\mathbb P}}(W=0)=\lim _{s\rightarrow \infty }\Phi _{W}(s)=a_1\). We then obtain \({{\mathbb E}}(W)=(1-a_1)/\mu \) in accordance with (42).
$$\begin{aligned} a_0(r)= \frac{r\mu }{1-r}, \quad a_1(r)= \frac{r}{1-r}\left( 1-\frac{(1+a)r\Phi _{A}(\mu )}{1 +a+ ar\Phi _{A}(\mu )}\right) -\frac{(1+a)r\Phi _{A}(\mu )}{1 +a+ ar\Phi _{A}(\mu )}e^{-a\mu w}. \end{aligned}$$
$$\begin{aligned} a_0=\mu ,\quad a_1=1-\frac{(1+a)\,\Phi _{A}(\mu )}{1 +a+ a\,\Phi _{A}(\mu )}, \end{aligned}$$
$$\begin{aligned} \Phi _{W}(s) = \frac{a_0+a_1s}{\mu +s} ={{\mathbb P}}(W>0)\frac{\mu }{\mu +s}+ {{\mathbb P}}(W=0), \end{aligned}$$
(43)
The case where \(a=1\), yielding the Lindley-type recursion \(W_{i+1} = \left[ B_i-A_i-W_i\right] ^+\), has been extensively studied in [23]. We obtain for the stationary processwhich is in agreement with [23, Formula (4.12), p. 74]. It is easy to see that \({{\mathbb P}}(W=0)=a_1\) is increasing in a.
$$\begin{aligned} a_0= \mu , \quad a_1= \frac{2-\Phi _{A}(\mu )}{2+ \Phi _{A}(\mu )}, \end{aligned}$$
For \(a=0\), we have \({{\mathbb P}}(W=0) = 1 -\Phi _{A}(\mu )\). This relation is explained by observing that now \({{\mathbb P}}(W=0) = {{\mathbb P}}(B<A)\), with \(B \sim \mathrm{exp}(\mu )\). For \(a \uparrow \infty \) we have \({{\mathbb P}}(W=0) = 1/(1+\Phi _{A}(\mu ))\), which is explained by observing that a positive W is followed by a geometric (q) number of zeroes, with \(q={{\mathbb P}}(B<A) = 1 -\Phi _{A}(\mu )\).
5 Model III: the uniform proportional case
In this section, we once more consider the stochastic recursion \(W_{i+1} = \left[ V_{i} W_i + Y_i\right] ^+\), where \(Y_i=B_i-A_i\). Again we impose the usual independence assumptions on the sequences \(\{V_i\}_{i\in {\mathbb N}_0}\), \(\{A_i\}_{i\in {\mathbb N}_0}\), and \(\{B_i\}_{i\in {\mathbb N}_0}\). In addition, we assume that the \(A_i\) are exp(\(\lambda \)) distributed. The ‘multiplicative adjustments’ \(\{V_i\}_{i\in {\mathbb N}_0}\) are assumed to form a sequence of unit uniformly distributed random variables on [0, 1]. By Theorem 2, since \({{\mathbb E}}(\log |V|)<0\), a steady-state distribution of \(\{W_i\}_{i\in {\mathbb N}_0}\) always exists. We shall first study its transient distribution and then obtain the steady-state distribution.
We start with (8), i.e.,where as before \(W^*_i=\left[ V_iW_i+B_i-A_i\right] ^-\). This time the distribution of \(W^*_i\) is almost trivial: either \(V_iW_i+B_i-A_i\geqslant 0\), in which case \(W^*_i=0\), or \(W^*_i\) has the same exponential distribution as \(A_i\), due to the lack of memory property of the exponential distribution.
$$\begin{aligned} \Phi _{W_{i+1}}(s) =\Phi _{V_iW_i+B_i-A_i}(s) +1-\Phi _{W^*_i}(s), \quad i=0,1,\dots , \end{aligned}$$
Using the independence between \(\{A_i\}_{i\in {\mathbb N}_0}\), \(\{B_i\}_{i\in {\mathbb N}_0}\), and \(\{V_i\}_{i\in {\mathbb N}_0}\) and the exponentiality of the \(A_i\), we obtainwhere we set \(p_i:= {{\mathbb P}}(W_{i}=0)\). Our goal is to write (44) fully in terms of the functions \(\Phi _{W_i}(s)\). To this end, performing the change of variable \(v:=su\), we obtainBy multiplying with \(\lambda -s\), we thus obtain the following recursive integral equation.
$$\begin{aligned} \Phi _{W_{i+1}}(s)&= \Phi _{V_iW_i}(s)\Phi _{B}(s)\frac{\lambda }{\lambda -s} + 1- {{\mathbb P}}(V_{i}W_{i}+B_i-A_i<0) \frac{\lambda }{\lambda -s}\nonumber \\&\quad -{{\mathbb P}}(V_{i}W_{i}+B_i-A_i\geqslant 0) \nonumber \\&= \Phi _{V_iW_i}(s)\Phi _{B}(s)\frac{\lambda }{\lambda -s} - p_{i+1} \frac{s}{\lambda -s}, \end{aligned}$$
(44)
$$\begin{aligned} \Phi _{V_iW_i}(s) = \int _0^1 {{\mathbb E}}(\mathrm{e}^{-suW_i})\,\mathrm{d}u=\frac{1}{s} \int _0^s \Phi _{W_i}(v)\,\mathrm{d}v. \end{aligned}$$
(45)
Lemma 7
For \(i\in {{\mathbb {N}}}\),
$$\begin{aligned} \Phi _{W_{i+1}}(s) = \frac{\lambda \,\Phi _{B}(s)}{s(\lambda -s)} \int _0^{s} \Phi _{W_i}(v)\,\mathrm{d}v-\frac{s}{\lambda -s}p_{i+1}. \end{aligned}$$
(46)
Since the LST \(\Phi _{W_0}(s)=\mathrm{e}^{-sw}\) of \(W_0\) is known, Relation (46) in principle allows us to recursively determine all the transforms \(\Phi _{W_i}(\cdot )\), \(i\in {{\mathbb {N}}}\). Observe that, when \(s=\lambda \), the right-hand side should become zero; using (45) we obtainThis formula can easily be interpreted probabilistically, using the memoryless property of the exponential distribution for \(A_i\):It is not possible to obtain explicit expressions for \(\Phi _{W_i}\). However, as so often, one can utilize the method of generating functions to turn the recursion (46) into some sort of differential or integral equation. Therefore, we multiply Eq. (46) by \(r^{i+1}\) and sum over i to obtainwhere, to simplify the notation,With \(I(s)=\int _0^{s}U_{W}(r,v)\,\mathrm{d}v\), we obtain the linear first-order differential equationAs follows by standard techniques, this inhomogeneous differential equation is solved bywhere necessarily \(\theta (c)=I(c)\) and we assume for the time being that \(c\in (s,\lambda )\) if \(s<\lambda \) and \(c\in (\lambda ,s)\) if \(\lambda <s\). Since \(\Phi _{B}\) is bounded and bounded away from zero, we have as \(s\downarrow \lambda \) (and likewise if \(s\uparrow \lambda \) in the case where \(s<\lambda \)),As a consequence, to make sure \(U_{W}(r,\lambda )\) remains bounded,and thereforeIt is to be noted that the integrand in (52) tends to \(\infty \) as \(s\rightarrow \lambda \), which follows from the finiteness of \(I(\lambda )\). Inserting (52) into (50), and recalling that \(U_{W}(r,s)=I'(s)\), yields the following expression for the generating function \(U_{W}\):Plugging (49) into (53) we obtainIt remains to determine \(U_{W}(r,\infty )\). Keeping in mind that \(U_{W}(r,s)-\kappa (s)\rightarrow 0\) by (49) we obtain, after some rearrangements,We summarize our findings in the following theorem.
$$\begin{aligned} p_{i+1} = \frac{\Phi _{B}(\lambda )}{\lambda } \int _0^{\lambda } \Phi _{W_i}(v)\,\mathrm{d}v. \end{aligned}$$
(47)
$$\begin{aligned} {{\mathbb P}}(W_{i+1}=0) = P(A_i \geqslant B_i + V_iW_i) = {{\mathbb P}}(A_i \geqslant B_i) P(A_i \geqslant V_iW_i) = \Phi _{B}(\lambda ) \Phi _{V_iW_i}(\lambda ). \end{aligned}$$
$$\begin{aligned} U_{W}(r,s)= \frac{\lambda r\,\Phi _{B}(s)}{s(\lambda -s)} \int _0^{s} U_{W}(r,v)\,\mathrm{d}v+\kappa (s), \end{aligned}$$
(48)
$$\begin{aligned} \kappa (s)=\Phi _{W_0}(s)-\frac{s}{\lambda -s}\left( U_{W}(r,\infty )-p_0\right) . \end{aligned}$$
(49)
$$\begin{aligned} I'(s) = \frac{\lambda r\Phi _{B}(s)}{s(\lambda -s)}I(s)+\kappa (s). \end{aligned}$$
(50)
$$\begin{aligned} I(s)=\mathrm{exp}\left( \lambda r \int _c^s\frac{\Phi _{B}(t)}{t(\lambda -t)}\,\mathrm{d}t \right) \left( \theta (c)+\int _c^s \kappa (u)\, \mathrm{exp}\left( -\lambda r\int _c^u\frac{\Phi _{B}(t)}{t(\lambda -t)}\,\mathrm{d}t \right) \,\mathrm{d}u\right) , \end{aligned}$$
(51)
$$\begin{aligned} \mathrm{exp}\left( \lambda r \int _c^s\frac{\Phi _{B}(t)}{t(\lambda -t)}\,\mathrm{d}t \right) \rightarrow \infty . \end{aligned}$$
$$\begin{aligned} \theta (c)=-\int _c^\lambda \kappa (u)\, \mathrm{exp}\left( -\lambda r\int _c^u\frac{\Phi _{B}(t)}{t(\lambda -t)}\,\mathrm{d}t \right) \,\mathrm{d}u, \end{aligned}$$
$$\begin{aligned} I(s)&=\mathrm{exp}\left( \lambda r \int _c^s\frac{\Phi _{B}(t)}{t(\lambda -t)}\,\mathrm{d}t \right) \int _\lambda ^s \kappa (u)\, \mathrm{exp}\left( -\lambda r\int _c^u\frac{\Phi _{B}(t)}{t(\lambda -t)}\,\mathrm{d}t \right) \,\mathrm{d}u \nonumber \\&=\int _\lambda ^s \kappa (u)\, \mathrm{exp}\left( \lambda r\int _u^s\frac{\Phi _{B}(t)}{t(\lambda -t)}\,\mathrm{d}t \right) \,\mathrm{d}u. \end{aligned}$$
(52)
$$\begin{aligned} U_{W}(r,s)=\kappa (s)+ \frac{\lambda r\Phi _{B}(s)}{s(\lambda -s)}\int _\lambda ^s \kappa (u)\, \mathrm{exp}\left( \lambda r\int _u^s\frac{\Phi _{B}(t)}{t(\lambda -t)}\,\mathrm{d}t \right) \,\mathrm{d}u. \end{aligned}$$
(53)
$$\begin{aligned} U_{W}(r,s)&=\kappa (s)+ \frac{\lambda r\Phi _{B}(s)}{s(\lambda -s)}\int _\lambda ^s \left( \Phi _{W_0}(u)-\frac{u\left( U_{W}(r,\infty )-p_0\right) }{\lambda -u}\right) \,\\&\quad \mathrm{exp}\left( \lambda r\int _u^s\frac{\Phi _{B}(t)}{t(\lambda -t)}\,\mathrm{d}t \right) \,\mathrm{d}u. \end{aligned}$$
$$\begin{aligned} U_{W}(r,\infty )=p_0-\frac{\int _\lambda ^\infty \Phi _{W_0}(u) \mathrm{exp}\left( -\lambda r\int _u^\infty \frac{\Phi _{B}(t)}{t(t-\lambda )}\,\mathrm{d}t \right) \,\mathrm{d}u}{\int _\lambda ^\infty \frac{u}{u-\lambda }\, \mathrm{exp}\left( -\lambda r\int _u^\infty \frac{\Phi _{B}(t)}{t(t-\lambda )}\,\mathrm{d}t \right) \,\mathrm{d}u}. \end{aligned}$$
Theorem 8
For \(r\in (0,1)\),where
$$\begin{aligned} U_{W}(r,s)=\kappa (s)+ \frac{\lambda r\Phi _{B}(s)}{s(\lambda -s)}\int _\lambda ^s \kappa (u)\, \mathrm{exp}\left( \lambda r\int _u^s\frac{\Phi _{B}(t)}{t(\lambda -t)}\,\mathrm{d}t \right) \,\mathrm{d}u, \end{aligned}$$
(54)
$$\begin{aligned} \kappa (s)=\Phi _{W_0}(s)+\frac{\frac{s}{\lambda -s}\int _\lambda ^\infty \Phi _{W_0}(u)\, \mathrm{exp}\left( -\lambda r\int _u^\infty \frac{\Phi _{B}(t)}{t(t-\lambda )}\,\mathrm{d}t \right) \,\mathrm{d}u}{\int _\lambda ^\infty \frac{u}{u-\lambda }\, \mathrm{exp}\left( -\lambda r\int _u^\infty \frac{\Phi _{B}(t)}{t(t-\lambda )}\,\mathrm{d}t \right) \,\mathrm{d}u}. \end{aligned}$$
The complexity of this type of result is comparable to that of recently studied related models; compare the structure of (54) with that of the transform of the transient storage level in, for example, [5].
We already noted that since the \(V_i\) are uniformly distributed on [0, 1] we have \({{\mathbb E}}(\log |V|)<0\). Hence, we always have \(W_i \Rightarrow W\) as \(i\rightarrow \infty \) for some proper random variable W. Its LST is given in the following theorem.
Theorem 9
\(W_i\) converges weakly to a proper limit W as \(i\rightarrow \infty \), andwhere
$$\begin{aligned} \Phi _{W}(s)=\frac{p_\infty }{s-\lambda }\left( s- \frac{\lambda \Phi _{B}(s)}{s}\int _\lambda ^s \frac{u}{u-\lambda }\, \mathrm{exp}\left( \lambda \int _u^s\frac{\Phi _{B}(t)}{t(\lambda -t)}\,\mathrm{d}t \right) \,\mathrm{d}u\right) , \end{aligned}$$
(55)
$$\begin{aligned} p_\infty = \left[ \int _0^\lambda \frac{1}{\lambda -u}\, \mathrm{exp}\left( -\int _0^u\left( \frac{\lambda \Phi _{B}(t)}{t(\lambda -t)}-\frac{1}{t} \right) \,\mathrm{d}t\right) \,\mathrm{d}u\right] ^{-1}. \end{aligned}$$
(56)
Proof
We apply an Abelian theorem and obtain (55) after multiplying both sides of (54) by \(1-r\) and letting r tend to one. We thereby use the fact thatThe relation for \(p_\infty \) follows by noting that \(\Phi _{W}(0)=1\), so thatNow,as \(t\downarrow 0\), so that we can safely let \(s\downarrow 0\) and obtain the finite and nonzero limit (56). \(\square \)
$$\begin{aligned} \lim _{r \uparrow 1} (1-r) \kappa (s)=\frac{sp_\infty }{s-\lambda }. \end{aligned}$$
$$\begin{aligned} \frac{1}{p_\infty }&=\lim _{s\downarrow 0} \frac{1}{s-\lambda }\left( s- \frac{\lambda \Phi _{B}(s)}{s}\int _\lambda ^s \frac{u}{u-\lambda }\, \mathrm{exp}\left( \lambda \int _u^s\frac{\Phi _{B}(t)}{t(\lambda -t)}\,\mathrm{d}t \right) \,\mathrm{d}u\right) \\&=\lim _{s\downarrow 0} \frac{1}{s}\int _s^\lambda \frac{u}{\lambda -u}\, \mathrm{exp}\left( -\lambda \int _s^u\frac{\Phi _{B}(t)}{t(\lambda -t)}\,\mathrm{d}t \right) \,\mathrm{d}u \\&=\lim _{s\downarrow 0} \int _s^\lambda \frac{1}{\lambda -u}\, \mathrm{exp}\left( -\int _s^u\left( \frac{\lambda \Phi _{B}(t)}{t(\lambda -t)}-\frac{1}{t} \right) \,\mathrm{d}t\right) \,\mathrm{d}u . \end{aligned}$$
$$\begin{aligned} \frac{\lambda \Phi _{B}(t)}{t(\lambda -t)}-\frac{1}{t}\rightarrow \frac{1}{\lambda }-{{\mathbb E}}(B), \end{aligned}$$
Remark 3
The expected value \({{\mathbb E}}(W)\) can be expressed in terms of the parameters \(\lambda \), \({{\mathbb E}}(B)\) and \(p_\infty \) as follows: Letting \(i\rightarrow \infty \) in (44) yieldsAfter a rearrangement of terms, this becomesAs \(s\downarrow 0\) the right-hand side tends to \(1+\lambda \left( {{\mathbb E}}(W)-{{\mathbb E}}(VW+B)\right) \) and since \({{\mathbb E}}(VW+B)={{\mathbb E}}(V){{\mathbb E}}(W)+{{\mathbb E}}(B)=\frac{1}{2}{{\mathbb E}}(W)+{{\mathbb E}}(B)\), we obtainThis yields the inequality \({{\mathbb P}}(W=0)\geqslant 1-\lambda {{\mathbb E}}(B)\), the value that one would get if \(V_i \equiv 1\).
$$\begin{aligned} \Phi _{W}(s) = \Phi _{VW}(s)\Phi _{B}(s)\frac{\lambda }{\lambda -s} - \frac{s}{\lambda -s}p_{\infty }. \end{aligned}$$
(57)
$$\begin{aligned} p_{\infty }=\Phi _{W}(s)+ \lambda \frac{1-\Phi _{W}(s)+\Phi _{VW+B}(s)-1}{s}. \end{aligned}$$
$$\begin{aligned} {{\mathbb E}}(W)=2\left( {{\mathbb E}}(B)-\frac{1-p_\infty }{\lambda }\right) . \end{aligned}$$
(58)
Example 2
If the \(B_i\) have a rational LST, Expression (55) for \(\Phi _W(s)\) and (56) for \(p_{\infty }\) simplify considerably. For exp(\(\mu \)) distributed \(B_i\), a partial fraction expansion yieldsFor \(s<\lambda \), the integral can be expressed in terms of the incomplete beta function \(B(x,a,b)=\int _0^x u^{a-1}(1-u)^{b-1}\,\mathrm{d}u\):We then obtainThis leads toso that, at least for \(s<\lambda \),Unfortunately, a similar expression for the \(s>\lambda \) case is not available. Instead, one obtains expressions that involve hypergeometric functions. Also it seems very hard to obtain higher moments from (59) by means of differentiation. It is possible, however, to derive a recursion formula for the moments \(\omega _k:={{\mathbb E}}(W^k)\) (where we assume their existence for \(k=1,2,\ldots ,j\), say) if we start with (57), which in our example becomesFor \(s<0\), the expansion \(\Phi _{W}(-s)=\sum _{k=0}^j \frac{\omega _k}{k!}s^k+o(s^j)\) holds. Inserting this into (60) yieldsEquating the coefficients on both sides leads toThen, in accordance with the general result (58),Moreover,and
$$\begin{aligned} \frac{u}{\lambda -u}\mathrm{exp}\left( \lambda \int _u^s\frac{\Phi _{B}(t)}{t(\lambda -t)}\,\mathrm{d}t \right) =\frac{s}{\lambda -u} \left( \frac{\lambda -u}{\lambda -s}\right) ^{\frac{\mu }{\lambda + \mu }} \left( \frac{\mu +u}{\mu +s}\right) ^{\frac{\lambda }{\lambda +\mu }}. \end{aligned}$$
$$\begin{aligned}&\int _\lambda ^s \frac{s}{u - \lambda } \left( \frac{\lambda -u}{\lambda -s}\right) ^{\frac{\mu }{\lambda + \mu }} \left( \frac{\mu +u}{\mu +s}\right) ^{\frac{\lambda }{\lambda +\mu }}\,\mathrm{d}u\\&\quad =\frac{s(\lambda +\mu )}{(\mu +s)^{\frac{\lambda }{\lambda +\mu }}(\lambda -s)^{\frac{\mu }{\lambda + \mu }}} B\left( \tfrac{\lambda -s}{\lambda +\mu },\tfrac{\mu }{\lambda +\mu },1+\tfrac{\lambda }{\lambda +\mu }\right) . \end{aligned}$$
$$\begin{aligned} \Phi _{W}(s)=p_\infty \cdot \left( \frac{\lambda \mu (\lambda +\mu )B\left( \tfrac{\lambda -s}{\lambda +\mu },\tfrac{\mu }{\lambda +\mu },1+\tfrac{\lambda }{\lambda +\mu }\right) }{(\mu +s)^{1+\frac{\lambda }{\lambda +\mu }}(\lambda -s)^{1+\frac{\mu }{\lambda + \mu }}} -\frac{s}{\lambda -s}\right) . \end{aligned}$$
$$\begin{aligned} p_\infty =\frac{\mu ^{\frac{\lambda }{\lambda +\mu }}\lambda ^{\frac{\mu }{\lambda + \mu }}}{(\lambda +\mu )B\left( \tfrac{\lambda }{\lambda +\mu },\tfrac{\mu }{\lambda +\mu },1+\tfrac{\lambda }{\lambda +\mu }\right) }, \end{aligned}$$
$$\begin{aligned} \Phi _{W}(s)&=\frac{\mu ^{\frac{\lambda }{\lambda +\mu }}\lambda ^{\frac{\mu }{\lambda + \mu }}}{(\lambda +\mu )B\left( \tfrac{\lambda }{\lambda +\mu },\tfrac{\mu }{\lambda +\mu },1+\tfrac{\lambda }{\lambda +\mu }\right) }\nonumber \\&\quad \cdot \left( \frac{\lambda \mu (\lambda +\mu )B\left( \tfrac{\lambda -s}{\lambda +\mu },\tfrac{\mu }{\lambda +\mu },1+\tfrac{\lambda }{\lambda +\mu }\right) }{(\mu +s)^{1+\frac{\lambda }{\lambda +\mu }}(\lambda -s)^{1+\frac{\mu }{\lambda + \mu }}} -\frac{s}{\lambda -s}\right) . \end{aligned}$$
(59)
$$\begin{aligned} (\lambda -s)\Phi _{W}(s) = \frac{\lambda \mu }{s(\mu +s)} \int _0^{s} \Phi _{W}(v)\,\mathrm{d}v-sp_{\infty }. \end{aligned}$$
(60)
$$\begin{aligned} (\mu -s)(\lambda +s)\sum _{k=0}^j \frac{\omega _k}{k!}s^k+o(s^j) = \lambda \mu \sum _{k=0}^j \frac{\omega _k}{(k+1)!}s^{k}-s(\mu +s)p_{\infty }+o(s^j), \quad s \downarrow 0. \end{aligned}$$
$$\begin{aligned}&\lambda \mu \frac{\omega _k}{k!}+ (\mu -\lambda )\frac{\omega _{k-1}}{(k-1)!}-\frac{\omega _{k-2}}{(k-2)!}{\mathbb {1}}_{\{k\geqslant 2\}} \\&=\left( \lambda \mu \frac{\omega _1}{2}-\mu p_\infty \right) {\mathbb {1}}_{\{k=1\}}+\left( \lambda \mu \frac{\omega _2}{6}-p_\infty \right) {\mathbb {1}}_{\{k=2\}}+ \frac{\lambda \mu \omega _k}{(k+1)!}{\mathbb {1}}_{\{k\geqslant 3\}} ,\quad k\in \{1,\ldots ,j\}. \end{aligned}$$
$$\begin{aligned} \omega _1= 2\left( \frac{1}{\mu }-\frac{1-p_\infty }{\lambda }\right) . \end{aligned}$$
$$\begin{aligned} \omega _2=3\frac{1-p_\infty -(\mu -\lambda )\omega _1}{\mu \lambda }, \end{aligned}$$
$$\begin{aligned} \omega _k =\frac{(k^2-1)\omega _{k-2}-(k+1)(\mu -\lambda )\omega _{k-1}}{\lambda \mu },\quad k\in \{3,\ldots ,j\}. \end{aligned}$$
Remark 4
One can generalize Model III to the case in which \(V=U^{1/\alpha }\), where U has a uniform distribution on [0, 1] and \(\alpha >0\). In this case (46) becomesLetting \(U_{W}^{(\alpha )}(r,s)=s^{\alpha -1}U_{W}(r,s)\) this yieldswhereEquation (61) is of the exact same type as (48), only with r replaced by \(r\alpha \). This allows one to derive \(U_{W}^{(\alpha )}(r,s)\) in the same way as before.
$$\begin{aligned} s^{\alpha -1}\Phi _{W_{i+1}}(s) = \frac{\alpha \lambda \,\Phi _{B}(s)}{s(\lambda -s)} \int _0^{s} v^{\alpha -1}\Phi _{W_i}(v)\,\mathrm{d}v-\frac{s^\alpha }{(\lambda -s)} p_{i+1}. \end{aligned}$$
$$\begin{aligned} U_{W}^{(\alpha )}(r,s)= \frac{\lambda r\alpha \,\Phi _{B}(s)}{s(\lambda -s)} \int _0^{s} U_{W}^{(\alpha )}(r,v)\,\mathrm{d}v+\kappa ^{(\alpha )}(s), \end{aligned}$$
(61)
$$\begin{aligned}\kappa ^{(\alpha )}(s)=s^{\alpha -1}\Phi _{W_0}(s)-\frac{s^\alpha }{\lambda -s}(U_{W}^{(\alpha )}(r,\infty )-p_0).\end{aligned}$$
6 Discussion and concluding remarks
This paper has analyzed three reflected (or delayed at zero) autoregressive processes specified by the stochastic recursion \(W_{i+1} = [V_iW_i + B_i - A_i]^+\). While the classical case of \(V\equiv 1\) has been widely studied in the queueing literature, our more general setting allows explicit analysis only in special cases. The three special cases we have considered are: (i) V equals a positive value a with certain probability \(p\in (0,1)\) and is negative otherwise, and both A and B have a rational LST, (ii) V attains negative values only and B has a rational LST, (iii) V is uniformly distributed on [0, 1], and A is exponentially distributed. In all three cases, we present transient and stationary results, where the transient results are in terms of the transform at a geometrically distributed epoch.
Cases which might allow explicit analysis are, for example: Another possible line of research concerns scaling limits and asymptotics. In particular, tail asymptotics seem to be within reach; in heavy-tailed cases these may be identified relying on a Tauberian approach. One also anticipates that, under particular scalings, an explicit analysis is possible. Specifically, one would expect that a diffusion analysis similar to the one presented in [7] can be performed.
1.
A combination of Models II and III, allowing V to be either negative or having a distribution as in Remark 4.
Acknowledgements
The research of Boxma and Mandjes is partly funded by the NWO Gravitation Programme NETWORKS (Grant Number 024.002.003) and an NWO Top Grant (Grant Number 613.001.352). The research of Palmowski is partially supported by Polish National Science Centre Grant No. 2018/29/B/ST1/00756, 2019-2022.
Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.