Strong Stability in an (R, s, S) Inventory Model

Boualem RABTA∗ and Djamil AISSANIL.A.M.O.S.

Laboratory of Modelization and Optimization of SystemsUniversity of Bejaia, 06000 (Algeria)E-mail : lamos bejaia@hotmail.com

Abstract

In this paper, we prove the applicability of the strong stability method to inven-tory models. Real life inventory problems are often very complicated and they areresolved only through approximations. Therefore, it is very important to justify theseapproximations and to estimate the resultant error. We study the strong stabilityin a periodic review inventory model with an (R, s, S) policy. After showing thestrong v−stability of the underlying Markov chain with respect to the perturbationof the demand distribution, we obtain quantitative stability estimates with an exactcomputation of constants.

Keywords : Inventory control, (R, s, S) policy, Markov chain, Perturbation,Strong Stability, Quantitative Estimates.

Introduction

When modelling practical problems, one may often replace a real system by another one

which is close to it in some sense but simpler in structure and/or components. This

approximation is necessary because real systems are generally very complicated, so their

analysis can not lead to analytical results or it leads to complicated results which are not

∗Corresponding author : Boualem RABTA, L.A.M.O.S., Laboratory of Modelization and Optimizationof Systems, University of Bejaia, 06000 (Algeria). Tel : (213) 34 21 08 00. Fax : (213) 34 21 51 88. E-mail: brabta@yahoo.fr.

1

useful in practice. Also, all model parameters are imprecisely known because they are

obtained by means of statistical methods. Such circumstances suggest to seek qualitative

properties of the real system, i.e., the manner in which the system is affected by the

changes in its parameters. These qualitative properties include invariance, monotonicity

and stability (robustness). It’s by means of qualitative properties that bounds can be

obtained mathematically and approximations can be made rigorously [18]. On the other

hand, in case of instability, a small perturbation may lead to at least a finite deviation [11].

The paper by F.W. Harris [8] was the first contribution to inventory research. Since,

thousands of papers were written about inventory problems and this field still a wide area

for research. for a recent contribution see for exemple [15]. Inventory models are the first

stochastic models for which monotonicity properties were proved (see [12, 20]). In 1969,

Boylan has proved a robustness theorem for inventory problems by showing that the solu-

tion of the optimal inventory equation depends continuously on its parameters including

the demand distribution [5]. Recently, Chen and Zheng (1997) in [6] have shown that the

inventory cost in an (R, s, S) model is relatively insensitive to the changes in D = S − s.

Also, Tee and Rossetti in [19] examine the robustness of a standard model of multi-echelon

inventory systems by developing a simulation model to explore the model’s ability to pre-

dict system performance for a two-echelon one-warehouse, multiple retailer system using

(R,Q) inventory policies under conditions that violate the model’s fundamental modeling

assumptions.

As many other stochastic systems, inventory systems can be entirely described by a

Markov chain. We will say that the inventory system (or model) is stable if the underly-

2

ing Markov chain is stable. For the investigation of the stability of Markov chains many

methods have been elaborated. Kalashnikov and Tsitsishvili [11] have proposed a method

inspired from Liapunov’s direct method for differential equations. Also, we can find in

the literature the weak convergence method (Stoyan [18]), metric method (Zolotarev [22]),

renewal method (Borovkov [3]), strong stability method (Aıssani and Kartashov [1]), uni-

form stability method (Ipsen and Meyer [10]),... Some of these methods allow us to obtain

quantitative estimates in addition to the qualitative affirmation of the continuity.

The strong stability method elaborated in the beginning of the 80’s is applicable to

all operations research models which can be represented by a Markov chain. It has been

applied to queueing systems (see for example [2]). According to this approach, we suppose

that the perturbation is small with respect to a certain norm. Such a strict condition

allow us to obtain better estimations on the characteristics of the perturbed chain. For

definitions and results on this method we suggest [14].

In this paper, we apply the strong stability method to the investigation of the stability in

an (R, s, S) periodic review inventory model and to the obtention of quantitative estimates.

In section 1, we present the model, define the underlying Markov chain and study its

transient and permanent regime. Notations are introduced in section 2. The main results

of this paper are given in sections 3, 4 and 5. After perturbing the demand probabilities,

we first prove the strong v−stability of the considered Markov chain (section 3), estimate

the deviation of its transition matrix (section 4) and then, we give an upper bound to the

approximation error (section 5). A numerical example is given in section 6 to illustrate the

use of the method. Definition and basic theorems of the strong stability method are given

3

in appendix.

1 The model

Consider the following single-item, single-echelon, inventory problem. Inventory level is

inspected every R time units. At the beginning of each period, the manager decides

whether he should order and if yes, how much to order. Suppose that the outside supplier

is perfectly reliable and that orders arrive immediately. During period n, n ≥ 1, total

demand is a random variable ξn. Assume that ξn, n ≥ 1, are independent and identically

distributed (i.i.d.) random variables with common probabilities given by

ak = P (ξ1 = k), k = 0, 1, 2, ...

So, ak is the probability to have a demand of k items during the period.

For such inventory problems, (R, s, S) policy is proved to be optimal (see [17, 9, 21]).

According to this control rule, if at review moment (tn = nR, n ≥ 1), the inventory level

Xn is below or equal to s (Order-Point) we order so much items to raise the inventory level

to S (Order-Up-To level). Thus, at the beginning of n + 1 period, the size of the order

is equal to Zn = S − Xn. Figure (1) illustrates the mechanism of this model. Note that

for this model, R, s and S are decision variables which values should be optimally chosen

through a suitable algorithm (see for example [21]).

Assume that the initial inventory level s < X0 ≤ S (if not, we raise it to S by ordering

sufficient quantity). The inventory level Xn+1 at the end of period n + 1 is given by :

Xn+1 =

{

(S − ξn+1)+ if Xn ≤ s,

(Xn − ξn+1)+ if Xn > s,

4

Figure 1: The inventory level in the (R, s, S) inventory system with zero lead time.

where (A)+ = max(A, 0).

The random variable Xn+1 depends only on Xn and ξn+1, where ξn+1 is independent

of n and the state of the system. Thus, X = {Xn, n ≥ 0} is a homogeneous Markov chain

with values in E = {0, 1, ..., S}.

1.1 Transition probabilities

Suppose that at date tn = nR, Xn = i.

1. If 0 ≤ i ≤ s, then Xn+1 = (S − ξn+1)+, and we have two cases :

for j = 0,

P (Xn+1 = 0|Xn = i) = P (S − ξn+1 ≤ 0) = P (ξn+1 ≥ S) =∞

∑

k=S

ak,

for 0 < j ≤ S,

P (Xn+1 = j|Xn = i) = P (S − ξn+1 = j) = P (ξn+1 = S − j) = aS−j.

5

2. If s < i ≤ S, then Xn+1 = (i − ξn+1)+, and we distinguish the following cases :

for j = 0,

P (Xn+1 = 0|Xn = i) = P (i − ξn+1 ≤ 0) = P (ξn+1 ≥ i) =∞

∑

k=i

ak,

for 0 < j ≤ i,

P (Xn+1 = j|Xn = i) = P (i − ξn+1 = j) = P (ξn+1 = i − j) = ai−j,

for j > i,

P (Xn+1 = j|Xn = i) = 0.

So, we have

Pij =

∞∑

k=S

ak if 0 ≤ i ≤ s and j = 0,

aS−j if 0 ≤ i ≤ s and 1 ≤ j ≤ S,∞∑

k=i

ak if s + 1 ≤ i ≤ S and j = 0,

ai−j if s + 1 ≤ i ≤ S and 1 ≤ j ≤ i,

0 if s + 1 ≤ i ≤ S and j ≥ i + 1.

Thus, the transition matrix of the Markov chain X is given by

P =

0 1 s s + 1 S

0∞∑

S

ak aS−1 · · · aS−s aS−s−1 · · · a0

1∞∑

S

ak aS−1 · · · aS−s aS−s−1 · · · a0

......

. . ....

.... . .

...

s∞∑

S

ak aS−1 · · · aS−s aS−s−1 · · · a0

s + 1∞∑

s+1

ak as · · · a1 a0 0 0

......

. . ....

.... . . 0

S∞∑

S

ak aS−1 · · · aS−s aS−s−1 · · · a0

6

The Markov chain X is irreducible and aperiodic. It has a unique stationary distribution

π = (π0, π1, ..., πS) given by the solution of

π.P = πS∑

i=0

πi = 1 .

Furthermore, this stationary distribution is independent from the initial inventory level

X0.

1.2 Stationary probabilities

Consider the Markov chain Y = {Yn, n ≥ 1} given by

Yn = {Items on-hand at the beginning of the period n (just after order Zn−1 delivery)} ,

with initial state Y1 = X0. So, we have

Xn = (Yn − ξn)+

and

Yn+1 =

{

S if Yn − ξn ≤ s,

Yn − ξn if Yn − ξn > s.

Also, we define the Markov chain V = {Vn, n ≥ 1} by

Vn = S − Yn

So, we have

Vn+1 =

{

0 if Vn + ξn ≥ S − s,

Vn + ξn if Vn + ξn < S − s.

or equivalently

Vn+1 = (Vn + ξn) 1I{Vn+ξn<S−s} (1)

7

V is a Markov chain with states in {0, 1, ..., S − s − 1}. If we denote by V∞ the random

variable having the stationary distribution of the Markov chain V then

V∞ = (V∞ + ξ1) 1I{V∞+ξ1<S−s} (2)

This relation implies that for every 0 ≤ v ≤ S − s − 1

FV (v) = P (V∞ ≤ v) = C + (FV ∗ Fξ)(v), (3)

where C = 1 − (FV ∗ Fξ)(S − s − 1) is a normalisation constant and Fξ(x) = P (ξ1 ≤ x) is

the cumulative function common to all ξn, n ≥ 1. Since (3) is a renewal type equation, it

follows (cf. [16]) that its uniquely determined solution is given by

FV (v) = C(1 + H(v)),

where H =∞∑

n=1

F n∗ξ is the renewal function associated to Fξ. The constant C can be

easily determined by the condition FV (S − s− 1) = 1, therefore we obtain that the unique

invariant distribution of the Markov chain V is given by

FV (v) =1 + H(v)

1 + H(S − s − 1), 0 ≤ v ≤ S − s − 1.

Now, we can deduce the stationary distribution of the Markov chain X from relation

X∞ = (S − (V∞ + ξ1))+

where X∞ is the random variable having the stationary distribution of the Markov chain

X.

8

2 Preliminary and notations

In this section, we introduce necessary notations adapted to our case, i.e., the case of

a finite discrete Markov chain. For a general framework see [14] and comments in the

appendix.

Let X = {Xn, n ≥ 0} be a homogeneous discrete Markov chain with values in a finite

space E = {0, 1, ..N}, given by a transition matrix P . Assume that X admits a unique

stationary vector π.

For a transition matrix P and a function f : E → R, we define

Pf(i) =∑

j∈E

Pijf(j), ∀i ∈ E,

and for a vector µ ∈ RN+1 we define

µf =∑

i∈E

µif(i).

Also, f ◦ µ is the matrix having the form

f(i)µi,∀i, j ∈ E.

We introduce in RN+1 the special family of norms

‖µ‖v =∑

i∈E

v(i)|µi|,∀µ ∈ RN+1, (4)

where v : E → R+ is a function bounded from bellow by a positive constant (not necessary

finite). So, the induced norms on space of matrices of size (N + 1)× (N + 1) will have the

following form

‖Q‖v = maxi∈E

(v(i))−1∑

j∈E

|Qij|v(j), (5)

9

Also, for a function f : E → R,

‖f‖v = maxi∈E

(v(i))−1 |f(i)|.

3 Strong v−stability of the (R, s, S) inventory model

Denote by Σ the considered inventory model. Let X = {Xn, n ≥ 0} be the Markov chain

where Xn is the on-hand inventory level at date tn = nR, n ≥ 1 and X0 is the initial

inventory level. Let Σ′ be another inventory model with the same structure but with

demands ξ′n having common probabilities

a′k = P (ξ′1 = k), k = 0, 1, ...

Let X ′ = {X ′n, n ≥ 0} be the Markov chain where X ′

n is the on-hand inventory level at

date tn = nR, n ≥ 1 in Σ′ and X ′0 = X0 is the initial inventory level. Denote by P and Q

the transition operators of the Markov chains X and X ′ respectively.

Definition 3.1. (cf. [1]) We say that the Markov chain X verifying ‖P‖v < ∞ is strongly

v−stable, if every stochastic matrix Q in the neighborhood {Q : ‖Q − P‖v < ε} admits a

unique stationary vector ν and :

‖ν − π‖v −→ 0 when ‖Q − P‖v −→ 0.

Theorem 3.1. In the (R, s, S) inventory model with instant replenishments, the Markov

chain X = {Xn, n ≥ 0} is strongly v−stable for a function v(k) = βk for all β > 1.

Proof. To prove the strong v−stability of the Markov chain X for a function v(k) = βk,

β > 1, we check the conditions of Theorem A.1.

10

So, we choose the measurable function

h(i) =

{

1 if 0 ≤ i ≤ s,

0 if s < i ≤ S,

and the measure α = (α0, α1, ..., αS), where

αj = P0j.

Thus, we can easily verify that

• πh =S∑

i=0

πih(i) =s

∑

i=0

πi > 0,

• α1I =S∑

j=0

P0j = 1,

• αh =S∑

j=0

P0jh(j) =s

∑

j=0

P0j =∞∑

i=S−s

ai > 0.

It is obvious that

Tij = Pij − h(i)αj =

{

0 if 0 ≤ i ≤ s,

Pij otherwise,

is a nonnegative matrix.

Now, we aim to show that there exists some constant ρ < 1 such that Tv(k) ≤ ρv(k) for

all k ∈ E.

Compute Tv(k) =S∑

j=0

Tkjv(j). If 0 ≤ k ≤ s, then

Tv(k) = 0.

If s < k ≤ S, then we have

Tv(k) =k

∑

j=0

Pkjβj =

∞∑

i=k

ai +k

∑

j=1

ak−jβj =

∞∑

i=k

ai +k−1∑

i=0

aiβk−i

11

=∞

∑

i=k

ai + βk

k−1∑

i=0

aiβ−i =

∞∑

i=k

ai

βk+

k−1∑

i=s+1

aiβ−i +

s∑

i=0

aiβ−i

βk

≤

∞∑

i=k

ai

βs+1+

k−1∑

i=s+1

ai

βs+1+

s∑

i=0

aiβ−i

βk =

∞∑

i=s+1

ai

βs+1+

s∑

i=0

aiβ−i

βk.

It suffices to take

ρ =

∞∑

i=s+1

ai

βs+1+

s∑

i=0

aiβ−i (6)

which is smaller then 1 for all β > 1.

Now, we verify that ‖P‖v < ∞

‖P‖v = maxk∈{0,1,..,S}

1

βk

S∑

j=0

Pkjβj = max(A,B)

where

A = maxk∈{0,1,..,s}

1

βk

S∑

j=0

Pkjβj = max

k∈{0,1,..,s}

1

βk

[

∞∑

i=S

ai +S

∑

j=1

aS−jβj

]

=∞

∑

i=S

ai +S

∑

j=1

aS−jβj

= 1 −

S−1∑

i=0

ai +S−1∑

i=0

aiβS−i = 1 +

S−1∑

i=0

ai

(

βS−i − 1)

and

B = maxk∈{s+1,..,S}

1

βk

S∑

j=0

Pkjβj = max

k∈{s+1,..,S}

1

βk

[

∞∑

i=k

ai +k

∑

j=1

ak−jβj

]

= maxk∈{s+1,..,S}

1

βk

[

1 +k−1∑

i=0

ai

(

βk−i − 1)

]

≤ maxk∈{s+1,..,S}

1

βs+1

[

1 +k−1∑

i=0

ai

(

βk−i − 1)

]

≤1

βs+1

[

1 +S−1∑

i=0

ai

(

βS−i − 1)

]

≤ 1 +S−1∑

i=0

ai

(

βS−i − 1)

= A.

So, we have

‖P‖v = 1 +S−1∑

i=0

ai

(

βS−i − 1)

< ∞.

12

�

4 Deviation of the transition matrix

The considered inventory system is strongly v−stable. This means that its characteristics

can approximate those of a similar inventory model having different demand distribution

under the condition that this distribution is close (in some sense) to the ideal system

demand distribution. To characterize this proximity we define the quantity

W =∞

∑

i=S

|ai − a′i| +

S−1∑

i=0

|ai − a′i| β

S−i. (7)

Before estimating numerically the deviation between stationary distributions of the chains

X and X ′, we estimate the deviation of the transition matrix P of the chain X. So, we

have

Lemma 4.1. Let P (resp. Q) be the transition matrix of the Markov chain X (resp. X ′).

Then,

‖P − Q‖v ≤ W

Proof.

‖P − Q‖v = maxk∈{0,1,..,S}

1

βk

S∑

j=0

|Pkj − Qkj| βj = max(C,D)

where

C = maxk∈{0,1,..,s}

1

βk

S∑

j=0

|Pkj − Qkj| βj ≤ max

k∈{0,1,..,s}

1

βk

[

∞∑

i=S

|ai − a′i| +

S∑

j=1

∣

∣aS−j − a′S−j

∣

∣ βj

]

=∞

∑

i=S

|ai − a′i| +

S−1∑

i=0

|ai − a′i| β

S−i

13

and

D = maxk∈{s+1,..,S}

1

βk

S∑

j=0

|Pkj − Qkj| βj ≤ max

k∈{s+1,..,S}

1

βk

[

∞∑

i=k

|ai − a′i| +

k∑

j=1

∣

∣ak−j − a′k−j

∣

∣ βj

]

= maxk∈{s+1,..,S}

∞∑

i=k

|ai − a′i|

βk+

k−1∑

i=0

|ai − a′i| β

−i

= maxk∈{s+1,..,S}

∞∑

i=k

|ai − a′i|

βk+

k−1∑

i=s+1

|ai − a′i| β

−i +s

∑

i=0

|ai − a′i| β

−i

≤

∞∑

i=s+1

|ai − a′i|

βs+1+

s∑

i=0

|ai − a′i| β

−i

≤ W.

So, we obtain

‖P − Q‖v ≤∞

∑

i=S

|ai − a′i| +

S−1∑

i=0

|ai − a′i| β

S−i = W.

�

5 Stability inequalities

The purpose of this section is to obtain a numerical estimation of the deviation between

stationary distributions of the Markov chains X and X ′. This can be done using Theorem

A.2.

First, estimate ‖π‖v.

Lemma 5.1. Let γ be the constant given by

γ =

(

∞∑

i=S

ai +S−1∑

i=0

aiβS−i

)

(1 − ρ)(1 + H(S − s − 1))(8)

14

where ρ is given by (6) and H =∞∑

n=1

F n∗ξ is the renewal function associated to the cumulative

distribution function Fξ of the random variable ξ1. Then,

‖π‖v ≤ γ

Proof. According to Theorem A.2,

‖π‖v ≤ (αv)(1 − ρ)−1(πh).

Since

αv =S

∑

j=0

α(j)v(j) =S

∑

j=0

P0jβj =

∞∑

i=S

ai +S

∑

j=1

aS−jβj =

∞∑

i=S

ai +S−1∑

i=0

aiβS−i

and

πh =S

∑

i=0

πih(i) =s

∑

i=0

πi = 1 −

S∑

i=s+1

πi = 1 − P (X∞ > s) = 1 − P (S − (V∞ + ξ1) > s)

= 1−P (V∞+ξ1 ≤ S−s−1) = 1−(FV ∗Fξ)(S−s−1) = 1−H(S − s − 1)

1 + H(S − s − 1)=

1

1 + H(S − s − 1),

it follows that

‖π‖v ≤

(

∞∑

i=S

ai +S−1∑

i=0

aiβS−i

)

(1 − ρ)(1 + H(S − s − 1))= γ.

�

Now, we can prove the following result

Theorem 5.1. Let π and ν be the stationary distributions of the Markov chains X and

X ′ respectively. Then, under the condition

W <1 − ρ

1 + γ(9)

15

we have

‖π − ν‖v ≤Wγ (1 + γ)

1 − ρ − (1 + γ) W(10)

where ρ is given by (6) and γ by (8).

Proof. We have ‖π‖v ≤ γ and

‖1I‖v = maxk∈E

1

βk= 1.

Imposing the condition

W <1 − ρ

1 + γ

and replacing constants in (12) by their values, we obtain

‖π − ν‖v ≤Wγ (1 + γ)

1 − ρ − (1 + γ) W.

�

Remark 1. The quantitative estimates obtained by means of the strong stability method

and given by the last result gives an upper bound of the deviation of the stationary distri-

bution of the Markov chain X with respect to the perturbation of the transition matrix

‖π − ν‖ ≤ C(P )‖P − Q‖ (11)

Some other stability methods can also be used to obtain such a bound. For the ”Absolutely

stable chain” defined in [10], the survey of Cho and Mayer [7] collects and compares several

bounds of the form (11). Note that this method study the sensitivity of the individual

stationary probabilities of a finite Markov chain.

16

6 Numerical example

Consider the (R, s, S) inventory model with Poisson distributed demand. This can cor-

respond to the case where clients arrive following a Poisson distribution and request all

exactly one item. Now, we apply the above results to test the sensitivity of the model

when perturbing the arriving rate λ to λ′ = λ + ε (for example, ε can be the error when

estimating λ from real data). For model parameters we take R = 1, S = 6, s = 3, λ = 5.

The transition matrix P of the Markov chain X is constructed and computations are made

by means a computer program. In a first step, we vary β to see the impact of the choice

of the norm. The perturbed parameter we take is λ′ = 5, 1 so ε = 0, 1. For each value of

β, the program compute the deviation W of the transition matrix and check whether the

condition (9) of Theorem 5.1 is satisfied. If yes, the program compute the upper bound of

the approximation error from relation (10).

The transition matrix of the chain X is given by

P =

0, 384039 0, 175467 0, 175467 0, 140374 0, 084224 0, 033690 0, 0067380, 384039 0, 175467 0, 175467 0, 140374 0, 084224 0, 033690 0, 0067380, 384039 0, 175467 0, 175467 0, 140374 0, 084224 0, 033690 0, 0067380, 384039 0, 175467 0, 175467 0, 140374 0, 084224 0, 033690 0, 0067380, 734974 0, 140374 0, 084224 0, 033690 0, 006738 0, 000000 0, 0000000, 559507 0, 175467 0, 140374 0, 084224 0, 033690 0, 006738 0, 0000000, 384039 0, 175467 0, 175467 0, 140374 0, 084224 0, 033690 0, 006738

The stationary distribution of the chain X is given by

π = (0, 416287, 0, 172774, 0, 167402, 0, 130486, 0, 076747, 0, 030288, 0, 006017)

Table 1 gives the results of the computations and the error e = ‖π − ν‖v = f(β) is

graphically represented in Figure 2. Observe that the condition (9) is satisfied only for

17

β W ‖π − ν‖v β W ‖π − ν‖v β W ‖π − ν‖v

1,11 0,042886 0,108725 1,21 0,052256 0,053748 1,31 0,064310 0,0492351,41 0,079634 0,057096 1,51 0,098904 0,075016 1,61 0,122900 0,1067491,71 0,152513 0,161400 1,81 0,188755 0,258115 1,91 0,232772 0,4406082,01 0,285853 0,828827 2,11 0,349444 1,894640 2,21 0,425156 9,022420

Table 1: The deviation of the transition matrix and the error e = ‖π − ν‖v = f(β) fordifferent values of β.

Figure 2: The error e = ‖π − ν‖v = f(β).

1, 03 ≤ β ≤ 2, 24. So, the choice of the norm (β) is relevant. The minimum estimation of

the error e = 0, 0489 is taken for β = 1, 29.

Now, let us vary the perturbation ε for a fixed value of β. The error e = ‖π−ν‖v = g(ε)

ε 0,01 0,02 0,03 0,04 0,1 0,2 0,3 0,4 0,5e 0,0043 0,0087 0,0133 0,0179 0,0489 0,1150 0,2091 0,3537 0,6041

Table 2: The error e = ‖π − ν‖v = g(ε).

is given in Table 2 for β = 1, 29. Naturally, the error is proportional to the perturbation

18

and the two quantities grow in the same time. Observe that the model have resisted to an

important perturbation (0, 5 represents 10% of the value of λ).

7 Conclusion

The Markov chain X denoting the on-hand inventory level at the end of periods in the

(R, s, S) inventory system with instant replenishments, is strongly v−stable with respect

to the perturbation of the demand distribution, for a function v(k) = βk, ∀β > 1. This

means that its characteristics are close to those of another inventory model with the same

structure but with demand distribution close to the ideal model demand distribution. The

last result gives us a numerical estimation of the error due to the approximation.

References

[1] Aıssani, D. and N.V. Kartashov, 1983, Ergodicity and stability of Markov chains with

respect to operator topology in the space of transition kernels, Dokl. Akad. Nauk.

Ukr. SSR, ser. A 11, 3–5.

[2] Aıssani, D. and N.V. Kartashov, 1984, Strong stability of the imbedded markov chain

in an M/G/1 system, International Journal Theor. Probability and Math. Statist. 29,

(American Mathematical Society), 1–5.

[3] Borovkov, A.A., 1972, Processus probabilistes de la theorie des files d’attente (Edition

Navka, Moskou).

19

[4] Boualouche, L. and D. Aıssani, 2002, Performance evaluation of an SW Communica-

tion Protocol (Send and Wait), Proceedings of the MCQT’02 (First Madrid Interna-

tional Conference on Queueing Theory), Madrid (Spain), 18.

[5] Boylan, E.S., 1969, Stability theorems for solutions to the optimal inventory equation,

Journal of Applied Probability 6, 211–217.

[6] Chen, F. and Y.S. Zheng, 1997, Sensitivity analysis of an (s, S) inventory model,

Operations Research Letters 21(1), 19–23.

[7] Cho, G. and C. Mayer, 2001, Comparaison of perturbation bounds for the stationary

probabilities of a finite Markov chain, Linear Algebra and its Applications, 335, 137–

150.

[8] Harris, F.W., 1913, How many parts to make at once, Factory (The magazine of

Management), 10(2), 135–136.

[9] Iglehart, D., 1963, Dynamic programming and stationary analysis of inventory prob-

lems, multistage inventory models and techniques, in : H. Scarf, D. Gilford, and

M. Shelly, editors, Multistage Inventory Models and Techniques, (Stanford University

Press) 1–31.

[10] Ipsen, I. and C. Meyer, 1994, Uniform stability of Markov chains, SIAM Journal on

Matrix Analysis and Applications, 15(4), 1061–1074.

20

[11] Kalashnikov, V.V. and G.Sh. Tsitsishvili, 1971, On the stability of queueing systems

with respect to disturbances of their distribution functions, Queueing theory and

reliability, 211–217.

[12] Karlin, S., 1960, Dynamic inventory policy with varing stochastic demands, Manage-

ment Sciences 6, 231–258.

[13] Kartashov, N.V., 1981, Strong Stability of Markov Chains, VNISSI, Vsesayouzni Sem-

inar on Stability Problems for stochastic Models, Moscow, 54-59 (See also : 1986, J.

Soviet Mat. 34, 1493–1498).

[14] Kartashov, N.V., 1996, Strong Stable Markov Chains (VSP, Utrecht, TbiMC Scientific

Publishers).

[15] Mohebbi, E., 2004, A replenishment model for the supply-uncertainty problem, Inter-

national Journal of Production Economics, 87(1), 25–37.

[16] Ross, S.M., 1970, Applied probability with optimization applications (Holden-Day,

San Francisco).

[17] Scarf, H., 1960, The optimality of (S, s) policies in the dynamic inventory problem, in

: K.J. Arrow, S. Karlin, and P. Suppes, editors, Mathematical methods in the social

sciences, (Stanford University Press, Stanford, Calif.) 196–202.

[18] Stoyan, D., 1983, Comparaison Methods for Queues and Other Stochastic Models,

English translation, D.J. Daley, Editor ( J. Wiley and Sons, New York).

21

[19] Y.S. Tee and M.D. Rossetti, 2002, A robustness study of a multi-echelon inventory

model via simulation, International Journal of Production Economics, 80(3), 265–277.

[20] Veinott, A., 1965, Optimal policy in a dynamic, single product, non-stationary inven-

tory model with several demand classes, Operations Research 13, 761–778.

[21] Veinott, A., and H. Wagner, 1965, Computing optimal (s, S) policies, Management

Sciences 11, 525–552.

[22] Zolotarev, V.M., 1975, On the continuity of stochastic sequenses generated by recurent

processes, Theory of Probability and Its Applications, XX(4), 819–832.

A Strong stability criterion

Conserve notations of section 2. The following theorem (cf. [1]) gives sufficient conditions

for the strong stability of a Harris recurrent Markov chain.

Theorem A.1. The Harris recurrent Markov chain X verifying ‖P‖v < ∞ is strongly

v-stable, if the following conditions are satisfied

1. ∃α ∈ RN+1+ ,∃h : E → R+ such that : πh > 0, α1I = 1, αh > 0,

2. T = P − h ◦ α is a nonnegative matrix,

3. ∃ ρ < 1 such that, Tv(x) ≤ ρ v(x),∀x ∈ E.

where 1I is the function identically equal to 1.

22

One important particularity of the strong stability method is the possibility to obtain

quantitative estimates. The following theorem (cf. [13]) allows us to obtain a numerical

estimation of the deviation of the stationary distribution of the strongly stable Markov

chain X.

Theorem A.2. Under conditions of theorem (A.1) and for ∆ verifying the condition

‖∆‖v < C−1(1 − ρ), we have :

‖ν − π‖v ≤ ‖∆‖v ‖π‖v C (1 − ρ − C ‖∆‖v)−1

, (12)

where

C = 1 + ‖ 1I ‖v ‖π‖v

and

‖π‖v ≤ (αv)(1 − ρ)−1(πh)

23