Modified Lindley Recursion

• Modified Lindley recursion is a family of stochastic recurrences that update waiting times by preserving nonnegativity through reflection.
• Variants include sign-reversed, Bernoulli-thinned, multiplicative, vector-valued, and distribution-theoretic models, each offering distinct analytical insights.
• Analytical methods such as fixed-point equations, Wiener–Hopf techniques, and contraction arguments are used to characterize stability, tail asymptotics, and recurrence.

Modified Lindley recursion denotes a family of stochastic recursions obtained by altering the classical Lindley waiting-time update while preserving the positive-part reflection that keeps the state nonnegative. In the works considered here, the modifications include a sign reversal in the feedback term, Bernoulli thinning of the recursion, multiplicative scaling of the current workload, vector-valued and ladder-epoch-embedded constructions, and a distinct distribution-theoretic recursion based on recursive mixtures. Across these formulations, the common analytic themes are fixed-point equations, fluctuation identities, Wiener–Hopf methods, contraction arguments, and recurrence or stationarity criteria for reflected processes (Cygan et al., 2017, Vlasiou, 2014, Boxma et al., 2022, Boxma et al., 2020).

1. Classical benchmark and scope of the modifications

The classical one-dimensional Lindley process arises in queueing theory as a reflected random walk. In the notation of the multidimensional recurrence study, it is

$$W_0=w_0\ge 0,\qquad W_n=\max\{W_{n-1}-Y_n,0\},\quad n\ge1,$$

and with $S_n=Y_1+\cdots+Y_n$, the process started at $W_0=0$ satisfies $W_n\overset{d}=M_n:=\max\{0,S_1,\dots,S_n\}$; moreover, its return times to $0$ coincide with the ascending ladder epochs of the underlying random walk (Cygan et al., 2017). This random-walk representation is the reference point from which the modified forms are best understood.

The literature covered here does not attach the phrase to a single canonical formula. Instead, several non-equivalent alterations of Lindley’s recursion are studied under closely related names. Some changes affect the sign of the feedback term, some randomize whether feedback is present at all, and others replace the additive memory term by multiplicative or vector-valued dependence.

Variant	Representative recursion	Distinguishing change
Classical benchmark	$W_{n+1}=[W_n+X_{n+1}]^+$	additive reflected walk
Sign-reversed	$W_{n+1}=\max\{0,B_{n+1}-A_n-W_n\}$	non-increasing in $W_n$
Bernoulli-thinned	$W\stackrel{d}{=}I(W+X)^+$	random suppression of feedback
Multiplicative	$W_{n+1}=[V_nW_n+Y_n]^+$	random scaling of memory
Vector / embedded	$S_n=Y_1+\cdots+Y_n$0 with ladder embeddings	recurrence via induced processes

A common misconception is that “modified Lindley recursion” refers only to the sign-reversed equation. The available works show a broader usage: the phrase also covers Bernoulli-thinned fluctuation recursions, multiplicative reflected AR(1)-type models, Markov-modulated vector recursions, and even a recursive distributional construction unrelated to workload dynamics (Vlasiou, 2014, Boxma et al., 2022, Dimitriou, 28 Aug 2025, Yaghoubi et al., 28 Dec 2025).

2. Sign-reversed and non-increasing Lindley-type equations

A prominent modification replaces the classical increasing dependence on the current waiting time by a decreasing one: $S_n=Y_1+\cdots+Y_n$1 or, in steady state,

$S_n=Y_1+\cdots+Y_n$2

Here $S_n=Y_1+\cdots+Y_n$3 are preparation times, $S_n=Y_1+\cdots+Y_n$4 are service times, and $S_n=Y_1+\cdots+Y_n$5 is the server waiting time in an alternating-service system with one server and two service points. Writing $S_n=Y_1+\cdots+Y_n$6, the recursion becomes $S_n=Y_1+\cdots+Y_n$7, and the map $S_n=Y_1+\cdots+Y_n$8 is decreasing in $S_n=Y_1+\cdots+Y_n$9, unlike the classical Lindley map (Vlasiou, 2014).

This sign reversal changes the analysis substantially. Standard monotonicity-based tools for the usual Lindley recursion do not apply directly, and the steady-state law must instead be characterized through a fixed-point equation for the distribution function $W_0=0$0: $W_0=0$1 Under $W_0=0$2, the process is regenerative with regeneration points at $W_0=0$3, is aperiodic, and has a unique stationary version. Analytically, the associated operator

$W_0=0$4

is a contraction on $W_0=0$5, with

$W_0=0$6

Banach’s fixed-point theorem then yields uniqueness, and the successive iteration $W_0=0$7 converges geometrically fast (Vlasiou, 2014).

The same paper studies tail asymptotics. If $W_0=0$8 is regularly varying with index $W_0=0$9, then

$W_n\overset{d}=M_n:=\max\{0,S_1,\dots,S_n\}$0

while if $W_n\overset{d}=M_n:=\max\{0,S_1,\dots,S_n\}$1 is rapidly varying, then

$W_n\overset{d}=M_n:=\max\{0,S_1,\dots,S_n\}$2

These asymptotics show that the waiting-time tail is governed by the tail of $W_n\overset{d}=M_n:=\max\{0,S_1,\dots,S_n\}$3, but modulated either by an exponential moment of $W_n\overset{d}=M_n:=\max\{0,S_1,\dots,S_n\}$4 or by the atom at zero, depending on the regime (Vlasiou, 2014).

The bounded-support study provides an exact solution when $W_n\overset{d}=M_n:=\max\{0,S_1,\dots,S_n\}$5 is exponential and $W_n\overset{d}=M_n:=\max\{0,S_1,\dots,S_n\}$6 has a polynomial distribution on $W_n\overset{d}=M_n:=\max\{0,S_1,\dots,S_n\}$7. Since $W_n\overset{d}=M_n:=\max\{0,S_1,\dots,S_n\}$8 is supported on $W_n\overset{d}=M_n:=\max\{0,S_1,\dots,S_n\}$9, the recursion forces $0$0 to lie in $0$1 as well. The fixed-point equation becomes an integral equation on a finite interval, which is converted into a high-order differential equation involving both $0$2 and $0$3, and then solved via a paired exponential ansatz. The limiting law has a point mass $0$4 at $0$5 and an absolutely continuous density on $0$6. For general bounded-support $0$7, Bernstein polynomial approximation yields a rigorous uniform bound

$0$8

where $0$9 (Vlasiou et al., 2014).

3. Multidimensional recursions, ladder epochs, and discrete subordination

In higher dimensions, the Lindley process is studied coordinatewise as

$W_{n+1}=[W_n+X_{n+1}]^+$0

but the recurrence analysis is not carried out solely through the coordinate recursion. Instead, it proceeds through ladder-time subordination, induced processes such as $W_{n+1}=[W_n+X_{n+1}]^+$1 and $W_{n+1}=[W_n+X_{n+1}]^+$2, and stochastic dynamical systems arguments. In this sense, the multidimensional process is treated as an embedded or modified Lindley-type system rather than merely a naive vector extension (Cygan et al., 2017).

For the classical one-dimensional process, the recurrence classification is inherited from the underlying random walk $W_{n+1}=[W_n+X_{n+1}]^+$3: the Lindley process is recurrent iff $W_{n+1}=[W_n+X_{n+1}]^+$4 is oscillating or has positive drift; it is null recurrent in the oscillating case and positive recurrent in the positive-drift case. Null recurrence occurs, for instance, when $W_{n+1}=[W_n+X_{n+1}]^+$5 or the increment law is symmetric, while positive recurrence occurs, for example, when $W_{n+1}=[W_n+X_{n+1}]^+$6 and $W_{n+1}=[W_n+X_{n+1}]^+$7 (Cygan et al., 2017).

The two-dimensional process $W_{n+1}=[W_n+X_{n+1}]^+$8 exhibits a more delicate dependence on fluctuation exponents. A key lemma states that if the first coordinate $W_{n+1}=[W_n+X_{n+1}]^+$9 and the projected process $W_{n+1}=\max\{0,B_{n+1}-A_n-W_n\}$0 are recurrent, then the full process is recurrent; the same implication holds for positive recurrence. Under independence of the coordinates, if both driving random walks are oscillating and their positivity parameters satisfy $W_{n+1}=\max\{0,B_{n+1}-A_n-W_n\}$1, then $W_{n+1}=\max\{0,B_{n+1}-A_n-W_n\}$2 is null recurrent. The same null recurrence conclusion holds when both coordinates are centered with finite second moment (Cygan et al., 2017).

The positivity parameter $W_{n+1}=\max\{0,B_{n+1}-A_n-W_n\}$3 enters through tail assumptions on the increment law. If $W_{n+1}=\max\{0,B_{n+1}-A_n-W_n\}$4, meaning that $W_{n+1}=\max\{0,B_{n+1}-A_n-W_n\}$5 belongs to the domain of attraction of a stable law with characteristic function

$W_{n+1}=\max\{0,B_{n+1}-A_n-W_n\}$6

then

$W_{n+1}=\max\{0,B_{n+1}-A_n-W_n\}$7

and Doney’s theorem gives $W_{n+1}=\max\{0,B_{n+1}-A_n-W_n\}$8. Equivalently, the first strict ascending ladder epoch $W_{n+1}=\max\{0,B_{n+1}-A_n-W_n\}$9 satisfies

$W_n$0

This ladder-epoch tail controls the induced dynamics (Cygan et al., 2017).

A central discrete subordination result considers an independent centered finite-range walk $W_n$1 on $W_n$2 observed at $W_n$3. The subordinated increment law has the asymptotic

$W_n$4

with

$W_n$5

and hence $W_n$6. This is the technical backbone of the mixed-process recurrence results for $W_n$7, where recurrence is reduced to that of $W_n$8 or $W_n$9 (Cygan et al., 2017).

When each coordinate process is positive recurrent, the $W\stackrel{d}{=}I(W+X)^+$0-dimensional recursion admits a unique invariant probability measure, almost sure convergence of the backward iterates, and a unique essential class. The proof uses local contractivity and stochastic dynamical systems methods rather than direct queueing arguments (Cygan et al., 2017).

4. Bernoulli-thinned recursions and fluctuation decompositions

A different modification arises in fluctuation theory for Lévy processes. The starting point is the deterministic recursion

$W\stackrel{d}{=}I(W+X)^+$1

which can be rewritten as the classical Lindley recursion

$W\stackrel{d}{=}I(W+X)^+$2

Under the relevant independence assumptions, the recursion solution is the running maximum of a random walk, and this observation is used to derive decomposition identities for the running maximum of a Lévy process and the time at which that maximum is last attained (Boxma et al., 2022).

The modified recursion introduced in this framework is the Bernoulli-thinned fixed-point equation

$W\stackrel{d}{=}I(W+X)^+$3

where $W\stackrel{d}{=}I(W+X)^+$4 is independent of $W\stackrel{d}{=}I(W+X)^+$5 and $W\stackrel{d}{=}I(W+X)^+$6. Proposition 2 identifies its unique solution as

$W\stackrel{d}{=}I(W+X)^+$7

where $W\stackrel{d}{=}I(W+X)^+$8 is the random walk with increments $W\stackrel{d}{=}I(W+X)^+$9 and $W_{n+1}=[V_nW_n+Y_n]^+$0 is geometric and independent of the walk. Corollary 6 extends this to

$W_{n+1}=[V_nW_n+Y_n]^+$1

whose unique solution is $W_{n+1}=[V_nW_n+Y_n]^+$2 with the same geometric stopping structure (Boxma et al., 2022).

This Bernoulli-thinned mechanism underlies a decomposition for a Lévy process $W_{n+1}=[V_nW_n+Y_n]^+$3 observed up to an exponential time $W_{n+1}=[V_nW_n+Y_n]^+$4. The one-dimensional identity writes $W_{n+1}=[V_nW_n+Y_n]^+$5 as the sum of two independent parts, one associated with killing rate $W_{n+1}=[V_nW_n+Y_n]^+$6 and one associated with Poisson$W_{n+1}=[V_nW_n+Y_n]^+$7 inspection. The two-dimensional extension applies to

$W_{n+1}=[V_nW_n+Y_n]^+$8

where $W_{n+1}=[V_nW_n+Y_n]^+$9 is the epoch at which the running maximum last occurs, and again decomposes the pair into two independent vector summands. The first is the continuously observed contribution up to the earlier-killed time $S_n=Y_1+\cdots+Y_n$00; the second is the contribution of a random walk formed by Lévy increments between inspection epochs, evaluated up to a geometric number of inspected intervals (Boxma et al., 2022).

The parameters $S_n=Y_1+\cdots+Y_n$01 and $S_n=Y_1+\cdots+Y_n$02 have distinct roles: $S_n=Y_1+\cdots+Y_n$03 is the killing rate and $S_n=Y_1+\cdots+Y_n$04 is the Poisson inspection rate. A striking feature is that $S_n=Y_1+\cdots+Y_n$05 appears only on the right-hand side of the decomposition. The explanation is the exponential-memoryless identity

$S_n=Y_1+\cdots+Y_n$06

with $S_n=Y_1+\cdots+Y_n$07. The paper emphasizes that this is the structural reason for the asymmetry in the formula, not an algebraic accident (Boxma et al., 2022).

Section 5 of the same work extends the mechanism to a two-dimensional generalized recursion that tracks both the maximum and its last attainment time: $S_n=Y_1+\cdots+Y_n$08 Its solution is the corresponding random-walk maximum and argmax pair,

$S_n=Y_1+\cdots+Y_n$09

where $S_n=Y_1+\cdots+Y_n$10 is the last time the random walk attains its maximum (Boxma et al., 2022).

5. Multiplicative, vector-valued, and Markov-modulated formulations

The multiplicative Lindley recursion replaces the unit coefficient in the classical additive update by a random multiplier: $S_n=Y_1+\cdots+Y_n$11 This reflected autoregressive process of order $S_n=Y_1+\cdots+Y_n$12 includes the classical Lindley recursion as the special case $S_n=Y_1+\cdots+Y_n$13. Writing $S_n=Y_1+\cdots+Y_n$14, with $S_n=Y_1+\cdots+Y_n$15 and $S_n=Y_1+\cdots+Y_n$16 independent i.i.d. nonnegative sequences and $S_n=Y_1+\cdots+Y_n$17 independent as well, the model is stable in particular if $S_n=Y_1+\cdots+Y_n$18 a.s. and $S_n=Y_1+\cdots+Y_n$19, or more simply if $S_n=Y_1+\cdots+Y_n$20 a.s. The explicit analysis in the paper treats three regimes: $S_n=Y_1+\cdots+Y_n$21 only with rational-LST $S_n=Y_1+\cdots+Y_n$22; mixed signs with $S_n=Y_1+\cdots+Y_n$23 with probability $S_n=Y_1+\cdots+Y_n$24 and negative otherwise, with rational-LST $S_n=Y_1+\cdots+Y_n$25 and $S_n=Y_1+\cdots+Y_n$26; and $S_n=Y_1+\cdots+Y_n$27 with exponential $S_n=Y_1+\cdots+Y_n$28, where the transform equation reduces to a solvable integral equation. In the uniform-multiplier case, the stationary mean satisfies

$S_n=Y_1+\cdots+Y_n$29

where $S_n=Y_1+\cdots+Y_n$30 (Boxma et al., 2020).

A more recent vector-valued extension is

$S_n=Y_1+\cdots+Y_n$31

where $S_n=Y_1+\cdots+Y_n$32 is an irreducible Markov chain on a finite state space and reflection at $S_n=Y_1+\cdots+Y_n$33 is componentwise. The associated transform equation has multiple recursive terms,

$S_n=Y_1+\cdots+Y_n$34

with $S_n=Y_1+\cdots+Y_n$35, $S_n=Y_1+\cdots+Y_n$36, and the mappings $S_n=Y_1+\cdots+Y_n$37 assumed commutative. This is a matrix/vector analogue of earlier scalar functional equations. Iteration of the functional equation generates compositions of the contractions $S_n=Y_1+\cdots+Y_n$38, and the analysis uses Liouville’s theorem and Wiener–Hopf boundary value theory to characterize the unknown vector terms and obtain iterative representations (Dimitriou, 2 Jul 2025).

The Markov-modulated dependency model develops

$S_n=Y_1+\cdots+Y_n$39

with a finite-state irreducible background chain $S_n=Y_1+\cdots+Y_n$40. In Model I,

$S_n=Y_1+\cdots+Y_n$41

with $S_n=Y_1+\cdots+Y_n$42, and $S_n=Y_1+\cdots+Y_n$43. In Model II,

$S_n=Y_1+\cdots+Y_n$44

and

$S_n=Y_1+\cdots+Y_n$45

The main object is the stationary transform vector

$S_n=Y_1+\cdots+Y_n$46

For Model I the stationary transform satisfies

$S_n=Y_1+\cdots+Y_n$47

leading to an explicit product-form series. For Model II the paper derives a different matrix transform equation, recursive steady-state moments, and exponential tail decay: if $S_n=Y_1+\cdots+Y_n$48 is the rightmost negative zero of $S_n=Y_1+\cdots+Y_n$49, then

$S_n=Y_1+\cdots+Y_n$50

for large $S_n=Y_1+\cdots+Y_n$51 (Dimitriou, 28 Aug 2025).

These multiplicative and Markov-modulated models show that one major sense of “modified” is replacement of additive memory by state-dependent multiplicative memory. The queueing interpretation is retained—waiting times and workloads remain the primary objects—but the transform analysis becomes genuinely matrix-valued and branch-recursive (Dimitriou, 2 Jul 2025, Dimitriou, 28 Aug 2025).

6. Exact solvable increment laws and distribution-theoretic usage

For the reflected random walk

$S_n=Y_1+\cdots+Y_n$52

with i.i.d. Laplace increments $S_n=Y_1+\cdots+Y_n$53, the process remains a Lindley recursion but the increment law allows exact finite-time analysis. The Laplace density is specified by location parameter $S_n=Y_1+\cdots+Y_n$54 and scale parameter $S_n=Y_1+\cdots+Y_n$55, with $S_n=Y_1+\cdots+Y_n$56 and $S_n=Y_1+\cdots+Y_n$57. The drift parameter determines the long-run regime: $S_n=Y_1+\cdots+Y_n$58 gives transience, $S_n=Y_1+\cdots+Y_n$59 gives null recurrence, and $S_n=Y_1+\cdots+Y_n$60 gives positive recurrence and convergence to a stationary distribution (Lucrezia et al., 2023).

The finite-time law of $S_n=Y_1+\cdots+Y_n$61 is mixed, consisting of a Dirac mass at $S_n=Y_1+\cdots+Y_n$62 and a continuous density on $S_n=Y_1+\cdots+Y_n$63. The paper derives recursive closed forms for the density $S_n=Y_1+\cdots+Y_n$64, with separate formulas for $S_n=Y_1+\cdots+Y_n$65, $S_n=Y_1+\cdots+Y_n$66, and $S_n=Y_1+\cdots+Y_n$67. It also analyzes the first exit time

$S_n=Y_1+\cdots+Y_n$68

from the strip $S_n=Y_1+\cdots+Y_n$69, obtaining explicit recursive formulas for $S_n=Y_1+\cdots+Y_n$70 in the cases $S_n=Y_1+\cdots+Y_n$71, $S_n=Y_1+\cdots+Y_n$72, and $S_n=Y_1+\cdots+Y_n$73, as well as simplified recursions when $S_n=Y_1+\cdots+Y_n$74 is small relative to $S_n=Y_1+\cdots+Y_n$75. This line of work shows that, for some increment laws, modified Lindley dynamics admit exact time-dependent distributions rather than only asymptotic characterizations (Lucrezia et al., 2023).

In a different and non-queueing sense, the term also appears in statistical distribution theory through a recursive construction combining Lindley-type and Gamma densities. The proposed recursion is

$S_n=Y_1+\cdots+Y_n$76

with

$S_n=Y_1+\cdots+Y_n$77

After $S_n=Y_1+\cdots+Y_n$78 iterations, the resulting law is the mixture

$S_n=Y_1+\cdots+Y_n$79

equivalently

$S_n=Y_1+\cdots+Y_n$80

Here $S_n=Y_1+\cdots+Y_n$81 is a recursion depth rather than a distribution parameter, and the paper fixes $S_n=Y_1+\cdots+Y_n$82 for the subsequent analysis (Yaghoubi et al., 28 Dec 2025).

This distribution-theoretic recursion has explicit survival and hazard functions,

$S_n=Y_1+\cdots+Y_n$83

$S_n=Y_1+\cdots+Y_n$84

raw moments

$S_n=Y_1+\cdots+Y_n$85

and moment generating function

$S_n=Y_1+\cdots+Y_n$86

Its hazard rate is decreasing for $S_n=Y_1+\cdots+Y_n$87, increasing for $S_n=Y_1+\cdots+Y_n$88, and bathtub-shaped for $S_n=Y_1+\cdots+Y_n$89. Maximum-likelihood estimation is carried out numerically via the Newton–Raphson method, and sums of i.i.d. variables from this law remain finite mixtures of Gamma distributions (Yaghoubi et al., 28 Dec 2025).

Taken together, these exact and recursive constructions show that modified Lindley recursion is not a single model class but a collection of analytically structured variants. The shared core is reflection through the positive part or recursive inheritance from a Lindley-type object; the substantive differences lie in whether the modification acts through sign reversal, thinning, multiplicative scaling, vector coupling, embedded ladder times, or recursive mixture formation (Lucrezia et al., 2023, Yaghoubi et al., 28 Dec 2025).