- Email: [email protected]
Transient solution of a random walk with chemical rule
ARTICLE IN PRESS
Physica A 382 (2007) 430–438 www.elsevier.com/locate/physa
Transient solution of a random walk with chemical rule A.M.K. Tarabia, A.H. El-Baz Department of Mathematics, Damietta Faculty of Science, New Damietta 34517, Egypt Received 12 March 2007; received in revised form 6 April 2007 Available online 19 April 2007
Abstract Conolly et al. [Math. Scientist 22 (1997) 83–91] have obtained the transient distribution for a random walk moving on the integers 1oko1 of the real line. Their analysis is based on a generating function technique. In this paper, an alternative technique is used to derive elegant explicit expressions for the transient state distribution of an infinite random walk having ‘‘chemical’’ rule and starting initially at any arbitrary integer position (say i). As a special case of our result, Conolly et al.’s (1997) solution is easily obtained. Moreover, the transient solution of the infinite symmetric continuous random walk is also presented. Finally, numerical values testing the quality of our analytical results are illustrated. r 2007 Elsevier B.V. All rights reserved. Keywords: Infinite chemical queue; Continuous random walk; Transient state; Difference equations
1. Introduction For over 50 years, determining new method to obtain the transient probability functions of the classical single-server queueing system has captured maintained the interest of theorists practitioners alike. Sharma and Bunday [1] and Sharma and Tarabia [2] have given a new simple approach to obtain the transient solution for the number of customers in the single-server queue with finite waiting space, which improved recently by Tarabia [3] and Tarabia and El-Baz [4]. Further simplifying this procedure, we present a similar approach to obtain the exact distribution for the transient state solution of Conolly et al. [5] model. In such model, a molecule can be modeled as infinitely long chain of atoms joined by links of equal length. The links are subjected to random chocks and this causes the atoms to move and the molecule to diffuse. This model can be described as queue of taxis and passengers at taxi stand. In addition, this model has widely physical applications with a similar probabilistic formalism arises in statistical mechanics in connection with the Ising model, for example. In the apparently extremely simplified cases discussed in Refs. [6,7]. Based on generating function and Laplace transform technique, Conolly et al. [5] derived the transient distribution of an unrestricted random walk initially starting at zero. The derived expressions are: ! 1 k 2kþ1 X X k þ 1 k ðmtÞ p2nþ1 ðtÞ ¼ eat r2rþ1n ; rþ1 ð2k þ 1Þ! r¼n r n k¼n
n ¼ 0; 1; 2; . . . ,
(1)
Corresponding author. Present address: Mathematics Department, Al-Jouf Faculty of Science, P.O. Box 2014, Sakaka, Saudi Arabia.
E-mail addresses: [email protected], [email protected] (A.M.K. Tarabia). 0378-4371/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2007.04.022
ARTICLE IN PRESS A.M.K. Tarabia, A.H. El-Baz / Physica A 382 (2007) 430–438
p2n ðtÞ ¼ e
at
1 k X k k ðmtÞ2k X r2k2rþn ; ð2kÞ! r n r r¼n k¼n
n ¼ 0; 1; 2; . . . ,
431
(2)
where a ¼ l þ m; r ¼ l=m. Adopting the procedures of Sharma and Tarabia [2] and Tarabia and El-Baz [4] for the transient state of Markovian queue this paper derives the transient probability of an unrestricted random walk starting initially at any position i, 1oio1. The procedure is based on converting the system of difference equations for the probabilities into a system of difference equations using a series form. The coefficients of this series satisfy simple recurrence relations, which permit the rapid and efficient evaluation of the transient state probabilities. A closed-form solution to these recurrence relations is obtained which leads to a simple formula for the transient state probabilities without the use of generating or Bessel functions. The rest of paper is organized as follows. In the next section, we describe the model and the notation used. Then, we consider a direct approach to obtain the solutions of Conolly et al. [5] model in elegant explicit formula. Moreover, we will show that Conolly et al. [5] solution is a special case of our result. The derived formula for the transient state is free of Bessel function or any integral forms. In Section 3, we will demonstrate that the transient solution of an infinite symmetric random walk on the real line is also a special case of our result. Finally, in Section 4 we gave a brief discussion based on numerical calculation to illustrate the derived theoretical results. 2. Formulation of the model and its analysis In this section, we describe the mathematical model in detail. The general process considered in this section can be described as follows. A molecule is modeled as an infinitely long chain of atoms joined by links of equal length. The links are subjected to random shocks and this causes the atoms to move and the molecule to diffuse. In the more interesting case considered here the atoms are of two alternating kinds and the shock mechanism is different according to whether the atom occupies an odd or an even position on the chain. It is easy to see that the given process can be described by a random walk {Q(t), tX0} with state space E ¼ f0; 1; 2; 3; . . .g where Q(t) denotes the atom position on the real line up to time t, with Q(0) ¼ i. Its transition probabilities satisfy the following ‘chemical rule’ for any integers k and l: 8 lh þ oðhÞ; k ¼ 2n þ 1; l ¼ 2n; > > > > < mh þ oðhÞ; k ¼ 2n 1; l ¼ 2n; probðQðt þ hÞ ¼ kjQðtÞ ¼ lÞ ¼ lh þ oðhÞ; k ¼ 2n; l ¼ 2n þ 1; > > > > : mh þ oðhÞ; k ¼ 2n; l ¼ 2n 1; where l and m are positive constants. Also let qi;n ðtÞ ¼ prob ðQðtÞ ¼ njQð0Þ ¼ iÞ;
i; n ¼ 0; 1; 2 . . .
or simply qn(t). The resulting stochastic behavior is described by a set of what are essentially Chapman–Kolmogorov forward equations. In the simplest case, these equations have a single parameter. From the above assumptions, imply that q_ 2n ðtÞ ¼ q2n1 ðtÞ ð1 þ rÞq2n ðtÞ þ rq2nþ1 ðtÞ,
(3)
q_ 2nþ1 ðtÞ ¼ rq2n ðtÞ ð1 þ rÞq2nþ1 ðtÞ þ q2nþ2 ðtÞ,
(4)
qn ð0Þ ¼ di;=n
(5)
with q_ n ðtÞ ¼ ðd=dtÞqn ðtÞ, di,n is the Kronecker’s delta, r ¼ l/m and the time has been scaled such that t ¼ mt. The following theorem without proof, called randomization or uniformization, is a well-known result used primarily for the numerical computation of the transition probability qn(t) of a Markov process (cf., Ref. [8,11]).
ARTICLE IN PRESS A.M.K. Tarabia, A.H. El-Baz / Physica A 382 (2007) 430–438
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Theorem 1. (Randomization) Suppose a Markov process (on a countable state space) has transition rate matrix P with SupiX0 pi;i pð1 þ rÞo1, then the transition probability qn(t) may be written as qn ðtÞ ¼ eð1þrÞt
1 X
PðmÞ i;n
m¼0
qn ðtÞ ¼ e
ð1þrÞt
ð1 þ rÞm tm ; m!
1 m X X m k ðmÞ r Pi;n k m¼0 k¼0
n ¼ 0; 1; 2; . . . , !
tm ; m!
n ¼ 0; 1; 2; . . . ,
where PðmÞ i;n is the m-step transition probability of the associated random Markov chain. Further, to make the recurrence Eqs. (3)–(5) simpler, the above equation can be rewritten as qn ðtÞ ¼ eð1þrÞt rni
1 X
cðm; n; iÞ
m¼0
tm ; m!
n ¼ 0; 1; 2; . . . .
(6)
It should be noted that qn(t) is completely determined once cðm; n; iÞ is known. Therefore, substituting from Eq. (6) into Eqs. (3)–(5), we get the following recurrence relations: cðm þ 1; 2n; iÞ ¼ r1 cðm; 2n 1; iÞ þ r2 cðm; 2n þ 1; iÞ; cðm þ 1; 2n þ 1; iÞ ¼ cðm; 2n; iÞ þ r cðm; 2n þ 2; iÞ;
n ¼ 0; 1; 2; . . . ,
n ¼ 0; 1; 2; . . . ,
(7) (8)
with cð0; n; iÞ ¼ din .
(9)
After a little algebra, it can be seen that cðm; n; iÞ is a polynomial in r and can be expressed as given in the following theorem. Theorem 2. For any integers n, i, and m, mX0, we get 8 0; m n þ i odd; > < ðmnþiÞ=2 ½ P cðm; n; iÞ ¼ > Bðm; n; rÞr2r ; m n þ i even; : r¼0
where 8 < A m; r; r þ niþ1 r½ðniÞ=2 ; 2 Bðm; n; rÞ ¼ : A m; r þ ni ; r r½ðniþ1Þ=2 ; 2 (i)
m ! Aðm; x; yÞ ¼
i
even;
i
odd:
mþ1 !
2
2
x
y
.
(ii) Að2m þ 1; x; yÞ ¼ Að2m; x; yÞ þ Að2m; x; y 1Þ. (iii) Að2m þ 2; x; yÞ ¼ Að2m þ 1; x; yÞ þ Að2m þ 1; x 1; yÞ. y Py whenever yox, or xo0 or xo0 and yo0. (iv) x . . . ¼ 0 ¼ x (v) [x] stands for the greatest integer not greater than x.
(10)
ARTICLE IN PRESS A.M.K. Tarabia, A.H. El-Baz / Physica A 382 (2007) 430–438
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Proof. We will carry out a complete induction on m. It can be easily verified that for simple cases m ¼ 0, 1, 2 formula (10) satisfies the recurrence relations (7)–(9). For example if i ¼ 1; n ¼ 0, and m ¼ 1, we have RHS ¼ cð1; 0; 1Þ þ rcð1; 2; 1Þ ¼ r2 þ r r1 ¼ 1 þ r2 ¼ cð2; 1; 1Þ ¼ LHS. Also, for i ¼ 2; n ¼ 1, and m ¼ 1, we get RHS ¼ r1 cð1; 1; 2Þ þ r2 cð1; 3; 2Þ ¼ r1 r þ r2 ð1Þ ¼ 1 þ r2 ¼ cð2; 2; 2Þ ¼ LHS:
&
In general, there are two possibilities viz either m is even or odd. Suppose m is even say 2k; we have the following cases: Case 1: If i is even say 2 l, we have from Eq. (7) RHS ¼ r1 cð2k; 2j 1; 2lÞ þ r2 cð2k; 2j þ 1; 2lÞ. Substituting from (10), we get RHS ¼ r1 :0 þ r2 :0 ¼ 0 ¼ cðm þ 1; 2j; iÞ. Similarly, from Eq. (8), we have RHS ¼ cð2k; 2j; 2lÞ þ rcð2k; 2j þ 2; 2lÞ ¼
kjþl X
Að2k; r; r þ j l Þr2rjþl þ
r¼0
kjþl1 X
Að2k; r; r þ j l þ 1Þr2rjþl1 :
r¼0
Using the fact that A(2k, kj+l, k+1) ¼ 0, which implies that RHS ¼
kjþl X
Að2k; r; r þ j l Þ þ Að2k; r; r þ j l þ 1Þ r2rjþl
r¼0
¼
kjþl X
Að2k þ 1; r; r þ j l þ 1Þr2rjþl
r¼0
¼ cðm þ 1; 2j þ 1; iÞ. Case 2: If i is odd say 2l+1, lX0, we have from Eq. (7) RHS ¼ r1 cð2k; 2j 1; 2l þ 1Þ þ r2 cð2k; 2j þ 1; 2l þ 1Þ ¼ r1
kjþlþ1 X
Að2k; r þ j l 1; rÞr2rjþlþ1 þ r2
r¼0
kjþl X
Að2k; r þ j l; rÞr2rjþl :
r¼0
Replacing r by r1 in the second term, we get RHS ¼
kjþlþ1 X
Að2k; r þ j l 1; rÞ þ Að2k; r þ j l 1; r 1Þ r2rjþl
r¼0
¼
kjþlþ1 X
Að2k þ 1; r þ j l 1; rÞr2rjþl
r¼0
¼ cðm þ 1; 2j; iÞ.
ARTICLE IN PRESS A.M.K. Tarabia, A.H. El-Baz / Physica A 382 (2007) 430–438
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Using Eq. (8), we get RHS ¼ cð2k; 2j; iÞ þ rcð2k; 2j þ 2; iÞ ¼
kjl1 X
Að2k; r; r þ j þ l þ 1Þr2rjþ3lþ2 þ r
kjl2 X
r¼0
Að2k; r; r þ j þ l þ 2Þr2rjþ3lþ1 .
r¼0
Apply the fact that Að2k; k j l 1; k þ 1Þ ¼ 0 to obtain RHS ¼
kjl1 X
Að2k; r; r þ j þ l þ 1Þ þ Að2k; r; r þ j þ l þ 2Þ r2rjþ3lþ2
r¼0
¼
kjl1 X
Að2k þ 1; r; r þ j þ l þ 2Þr2rjþ3lþ2
r¼0
¼ cðm þ 1; 2j þ 1; iÞ. Working likewise for the case m ¼ 2k+1 (say) and is omitted here for brevity. Hence, result (10) is fully established and thus, the solution of the differential-difference equations (3)–(5) can be written in the following cases: Case 1: If i is even say (2l) q2n ðtÞ ¼ eðlþmÞt
1 mnþl X X
Að2m; r; r þ n l Þr2rþnl
m¼0 r¼0
q2nþ1 ðtÞ ¼ eðlþmÞt
1 mnþl X X
ðmtÞ2m ; ð2mÞ!
n ¼ 0; 1; 2; . . . ,
(11)
Að2m þ 1; r; r þ n l þ 1Þr2rþnlþ1
ðmtÞ2mþ1 ; ð2m þ 1Þ!
n ¼ 0; 1; 2; . . . .
(12)
Að2m þ 1; r þ n l 1; rÞr2rþnl1
ðmtÞ2mþ1 ; ð2m þ 1Þ!
n ¼ 0; 1; 2; . . . ,
(13)
m¼0 r¼0
Case 2: If i is odd say (2l+1) q2n ðtÞ ¼ eðlþmÞt
1 mnþlþ1 X X m¼0
q2nþ1 ðtÞ ¼ eðlþmÞt
r¼0
1 mnþl X X
Að2m; r þ n l; rÞr2rþnl
m¼0 r¼0
ðmtÞ2m ; ð2mÞ!
n ¼ 0; 1; 2; . . . .
(14)
3. Special cases Case (a): Conolly et al. [5] formulae. In the following lemma, we will show that Eqs. (11) and (12) can be reduced to Conolly et al. [5] formulae. Lemma 1. Conolly et al. [5] formulae are a special case of our formulae (11) and (12) when i ¼ 0. Proof. Put i ¼ 0 in Eqs. (11) and (12), we obtain q2n ðtÞ ¼ eðlþmÞt
1 X mn X
Að2m; r; r þ nÞr2rþn
m¼0 r¼0
q2nþ1 ðtÞ ¼ eðlþmÞt
1 X mn X
ðmtÞ2m ; ð2mÞ!
n ¼ 0; 1; 2; . . . ,
Að2m þ 1; r; r þ n þ 1Þr2rþnþ1
m¼0 r¼0
ðmtÞ2mþ1 ; ð2m þ 1Þ!
n ¼ 0; 1; 2; . . . .
(15)
(16)
Replacing r by rn in Eqs. (15) and (16), we get q2n ðtÞ ¼ eðlþmÞt
1 X m X m¼0 r¼n
Að2m; r n; rÞr2rn
ðmtÞ2m , ð2mÞ!
(17)
ARTICLE IN PRESS A.M.K. Tarabia, A.H. El-Baz / Physica A 382 (2007) 430–438
q2nþ1 ðtÞ ¼ eðlþmÞt
1 X m X
Að2m þ 1; r n; r þ 1Þr2rnþ1
m¼0 r¼n
435
ðmtÞ2mþ1 . ð2m þ 1Þ!
(18)
Now, we have m X
Að2m; r n; rÞr
r¼n
m ¼
!
m
¼
! r þ
þ þ
m
r¼n
rn
!
m
mn2
m
m2
! r2rn
r
! r
nþ1 ! m
1 !
m
!
m X
m
n
n
0
2rn
nþ2
m þ
2mn4
r
!
!
m nþ2
2
m þ
mn1
rnþ4 !
!
m
r
m1
mn2
m þ
!
mn
m
!
m
r2mn .
Rearrangement the power of r and using the fact that m m ¼ , x mx we get m ¼
!
0
m n
! 2mn
r
m þ
m þ þ
mn2
!
!
1 m
!
m nþ1 !
m2
r
r
nþ4
2mn2
m þ
!
2
m þ
mn1
!
m
!
nþ2 m m1
r2mn4 ! r
nþ2
m þ
mn
!
m
!
m
rn
¼ Að2m; 0; nÞr2mn þ Að2m; 1; n þ 1Þr2mn2 þ Að2m; 2; n þ 2Þr2mn4 þ þ Að2m; m 2; m n 2Þrnþ4 þ Að2m; m 1; m n 1Þrnþ2 þ Að2m; m; m nÞrn m X ¼ Að2m; r n; rÞr2m2rþn . r¼n
Hence, Eq. (17) can be rewritten as follows: q2n ðtÞ ¼ eðlþmÞt
1 X m X m¼0 r¼n
Að2m; r n; rÞr2m2rþn
ðmtÞ2m . ð2mÞ!
(19)
Substituting by the values of A(2m,rn,r) and A(2m+1,rn,r+1) into Eqs. (19) and (18) respectively, Conolly et al. [5] formulae can be deduced. & Case (b): Symmetric random walk. If r ¼ 1 we will show that Eqs. (11)–(14) can be reduced to the coefficient of the well-known solution of unrestricted symmetric random walk. To achieve this, setting r ¼ 1 in formula (10) yields: Lemma 2. For any integers n, i, and m, mX0, and for r ¼ 1, we get 8 0; m n þ i odd; > > ! < m cðm; n; iÞ ¼ mnþi m n þ i even: > > : 2 Proof. Consider the case m ¼ 2k, n ¼ 2j, and i ¼ 2l, we get ! kjþl ½ðmnþiÞ=2 X X k k niþ1 cðm; n; iÞ ¼ . A m; r; r þ ¼ rþjl 2 r r¼0 r¼0
ARTICLE IN PRESS A.M.K. Tarabia, A.H. El-Baz / Physica A 382 (2007) 430–438
436
Using Riordan [9], we obtain !
2k
cðm; n; iÞ ¼
m
;
kjþl
!
mnþi .
¼
2
In the same manner, the proof of this proposition can be completed for the other cases. Hence, for any i, we have !
m
mnþi .
cðm; n; iÞ ¼
2
On account of the above proposition and if i ¼ 2l, we obtain q2n ðtÞ ¼ e2mt q2nþ1 ðtÞ ¼ e
2mt
!
1 X
2m
m¼0
mnþl
1 X
2m þ 1
m¼0
mnþl
!
ðmtÞ2m ; 2m!
n ¼ 0; 1; 2; . . . ,
ðmtÞ2mþ1 ; ð2m þ 1Þ!
n ¼ 0; 1; 2; . . . .
Similarly, if i ¼ 2l+1, we have q2n ðtÞ ¼ e
2mt
q2nþ1 ðtÞ ¼ e2mt
!
1 X
2m þ 1
m¼0
mnþl
1 X
2m þ 1
m¼0
mnþl
!
ðmtÞ2m ; 2m!
n ¼ 0; = 1; 2; . . . ,
ðmtÞ2mþ1 ; ð2m þ 1Þ!
n ¼ 0; 1; 2; . . . .
If i ¼ 0 and r ¼ 1, we have q0 ðtÞ ¼ e2mt
1 X 2m ðmtÞ2m m¼0
m
2m!
¼ e2mt I 0 ð2mtÞ,
1.2 q_8 Transient probabilities
1
q_6 q_4
0.8
q_2
0.6
q0 q2
0.4
q4 q6
0.2
q8 0 0
50
100
150
200
250
300
350
400
Values of time t Fig. 1. Transient probabilities of some even states when the system starting at i ¼ 2, l ¼ 0.1, m ¼ 0.2.
ARTICLE IN PRESS A.M.K. Tarabia, A.H. El-Baz / Physica A 382 (2007) 430–438
437
P 2m 2 where I 0 ð2mtÞ ¼ 1 m¼0 ððmtÞ =ðm!Þ Þ is a modified Bessel function (see Ref. [10]). Clearly, this agrees with Conolly et al. [5]. &
4. Numerical illustrations In this section, the computational results obtained by employing the above approach are discussed through three graphs. In Figs. 1 and 2 the transient probabilities of the even and odd states, respectively, are plotted for i ¼ 2, l ¼ 0.1, m ¼ 0.2. As can be seen, the transient behavior goes to zero for n-N and the given system has no steady-state values. In Fig. 3, we show the effects of the traffic intensity r on the transient probability q0(t) when the system is starting at i ¼ 2. Fig. 3 shows that for a given r, the transient probability q0(t) decreases as the parameter r decreases. This is understandable since for such values, the arrival rate is decreased as compared to the service rate, thus forcing the system to have more chance to reaches to zero as n-N.
0.3 q_7
Transient probabilities
0.25
q_5 0.2
q_3 q_1
0.15
q1 q3
0.1
q5 0.05 0
q7
0
100
200
300
400
500
Values of time t Fig. 2. Transient probabilities of some odd states when the system starting at i ¼ 2, l ¼ 0.1, m ¼ 0.2.
0.14 0.12
Probability
0.1
roh=0.1
0.08
roh=0.25 roh=0.5
0.06 0.04 0.02 0 0
100
200
300
400
Values of time t Fig. 3. Effects of the traffic intensity r on the transient probability q0(t) when the system starting at state i ¼ 2.
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Acknowledgments The authors are grateful to the Area Editor and the anonymous referees for their useful comments and suggestions, which helped to improve the earlier version of this paper. References [1] O.P. Sharma, D.B. Bunday, Optimization 40 (1997) 79–84. [2] O.P. Sharma, A.M.K. Tarabia, A simple transient analysis of an M/M/1/N queue, Sankhya¯ , Ser. A 62 (2000) 273–281. [3] A.M.K. Tarabia, A new formula for the transient behaviour of a non-empty M/M/1/N queue, Appl. Math. Comput. 132 (2002) 1–10. [4] A.M.K. Tarabia, A.H. El-Baz, Exact transient solutions of non-Markovian queues, Comput. Math. Appl. 52 (2006) 985–996. [5] B.W. Conolly, P.R. Parthasarathy, S. Dharmaraja, Math. Scientist 22 (1997) 83–91. [6] R.J. Glauber, J. Math. Phys. 4 (1963) 495–508. [7] W.H. Stockmayer, et al., Local-jump models for chain dynamics, Pure Appl. Chem. 26 (1971) 555–561. [8] D. Gross, C.M. Harris, Fundamentals of Queueing Theory, second ed, Wiley, New York, 1985. [9] J. Riordan, Combinatorial Identities, Wiley, New York, 1968. [10] M. Abromowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1970. [11] W.J. Anderson, Continuous-time Markov Chains. An Applications-oriented Approach, Springer, New York, 1991.
Physica A 382 (2007) 430–438 www.elsevier.com/locate/physa
Transient solution of a random walk with chemical rule A.M.K. Tarabia, A.H. El-Baz Department of Mathematics, Damietta Faculty of Science, New Damietta 34517, Egypt Received 12 March 2007; received in revised form 6 April 2007 Available online 19 April 2007
Abstract Conolly et al. [Math. Scientist 22 (1997) 83–91] have obtained the transient distribution for a random walk moving on the integers 1oko1 of the real line. Their analysis is based on a generating function technique. In this paper, an alternative technique is used to derive elegant explicit expressions for the transient state distribution of an infinite random walk having ‘‘chemical’’ rule and starting initially at any arbitrary integer position (say i). As a special case of our result, Conolly et al.’s (1997) solution is easily obtained. Moreover, the transient solution of the infinite symmetric continuous random walk is also presented. Finally, numerical values testing the quality of our analytical results are illustrated. r 2007 Elsevier B.V. All rights reserved. Keywords: Infinite chemical queue; Continuous random walk; Transient state; Difference equations
1. Introduction For over 50 years, determining new method to obtain the transient probability functions of the classical single-server queueing system has captured maintained the interest of theorists practitioners alike. Sharma and Bunday [1] and Sharma and Tarabia [2] have given a new simple approach to obtain the transient solution for the number of customers in the single-server queue with finite waiting space, which improved recently by Tarabia [3] and Tarabia and El-Baz [4]. Further simplifying this procedure, we present a similar approach to obtain the exact distribution for the transient state solution of Conolly et al. [5] model. In such model, a molecule can be modeled as infinitely long chain of atoms joined by links of equal length. The links are subjected to random chocks and this causes the atoms to move and the molecule to diffuse. This model can be described as queue of taxis and passengers at taxi stand. In addition, this model has widely physical applications with a similar probabilistic formalism arises in statistical mechanics in connection with the Ising model, for example. In the apparently extremely simplified cases discussed in Refs. [6,7]. Based on generating function and Laplace transform technique, Conolly et al. [5] derived the transient distribution of an unrestricted random walk initially starting at zero. The derived expressions are: ! 1 k 2kþ1 X X k þ 1 k ðmtÞ p2nþ1 ðtÞ ¼ eat r2rþ1n ; rþ1 ð2k þ 1Þ! r¼n r n k¼n
n ¼ 0; 1; 2; . . . ,
(1)
Corresponding author. Present address: Mathematics Department, Al-Jouf Faculty of Science, P.O. Box 2014, Sakaka, Saudi Arabia.
E-mail addresses: [email protected], [email protected] (A.M.K. Tarabia). 0378-4371/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2007.04.022
ARTICLE IN PRESS A.M.K. Tarabia, A.H. El-Baz / Physica A 382 (2007) 430–438
p2n ðtÞ ¼ e
at
1 k X k k ðmtÞ2k X r2k2rþn ; ð2kÞ! r n r r¼n k¼n
n ¼ 0; 1; 2; . . . ,
431
(2)
where a ¼ l þ m; r ¼ l=m. Adopting the procedures of Sharma and Tarabia [2] and Tarabia and El-Baz [4] for the transient state of Markovian queue this paper derives the transient probability of an unrestricted random walk starting initially at any position i, 1oio1. The procedure is based on converting the system of difference equations for the probabilities into a system of difference equations using a series form. The coefficients of this series satisfy simple recurrence relations, which permit the rapid and efficient evaluation of the transient state probabilities. A closed-form solution to these recurrence relations is obtained which leads to a simple formula for the transient state probabilities without the use of generating or Bessel functions. The rest of paper is organized as follows. In the next section, we describe the model and the notation used. Then, we consider a direct approach to obtain the solutions of Conolly et al. [5] model in elegant explicit formula. Moreover, we will show that Conolly et al. [5] solution is a special case of our result. The derived formula for the transient state is free of Bessel function or any integral forms. In Section 3, we will demonstrate that the transient solution of an infinite symmetric random walk on the real line is also a special case of our result. Finally, in Section 4 we gave a brief discussion based on numerical calculation to illustrate the derived theoretical results. 2. Formulation of the model and its analysis In this section, we describe the mathematical model in detail. The general process considered in this section can be described as follows. A molecule is modeled as an infinitely long chain of atoms joined by links of equal length. The links are subjected to random shocks and this causes the atoms to move and the molecule to diffuse. In the more interesting case considered here the atoms are of two alternating kinds and the shock mechanism is different according to whether the atom occupies an odd or an even position on the chain. It is easy to see that the given process can be described by a random walk {Q(t), tX0} with state space E ¼ f0; 1; 2; 3; . . .g where Q(t) denotes the atom position on the real line up to time t, with Q(0) ¼ i. Its transition probabilities satisfy the following ‘chemical rule’ for any integers k and l: 8 lh þ oðhÞ; k ¼ 2n þ 1; l ¼ 2n; > > > > < mh þ oðhÞ; k ¼ 2n 1; l ¼ 2n; probðQðt þ hÞ ¼ kjQðtÞ ¼ lÞ ¼ lh þ oðhÞ; k ¼ 2n; l ¼ 2n þ 1; > > > > : mh þ oðhÞ; k ¼ 2n; l ¼ 2n 1; where l and m are positive constants. Also let qi;n ðtÞ ¼ prob ðQðtÞ ¼ njQð0Þ ¼ iÞ;
i; n ¼ 0; 1; 2 . . .
or simply qn(t). The resulting stochastic behavior is described by a set of what are essentially Chapman–Kolmogorov forward equations. In the simplest case, these equations have a single parameter. From the above assumptions, imply that q_ 2n ðtÞ ¼ q2n1 ðtÞ ð1 þ rÞq2n ðtÞ þ rq2nþ1 ðtÞ,
(3)
q_ 2nþ1 ðtÞ ¼ rq2n ðtÞ ð1 þ rÞq2nþ1 ðtÞ þ q2nþ2 ðtÞ,
(4)
qn ð0Þ ¼ di;=n
(5)
with q_ n ðtÞ ¼ ðd=dtÞqn ðtÞ, di,n is the Kronecker’s delta, r ¼ l/m and the time has been scaled such that t ¼ mt. The following theorem without proof, called randomization or uniformization, is a well-known result used primarily for the numerical computation of the transition probability qn(t) of a Markov process (cf., Ref. [8,11]).
ARTICLE IN PRESS A.M.K. Tarabia, A.H. El-Baz / Physica A 382 (2007) 430–438
432
Theorem 1. (Randomization) Suppose a Markov process (on a countable state space) has transition rate matrix P with SupiX0 pi;i pð1 þ rÞo1, then the transition probability qn(t) may be written as qn ðtÞ ¼ eð1þrÞt
1 X
PðmÞ i;n
m¼0
qn ðtÞ ¼ e
ð1þrÞt
ð1 þ rÞm tm ; m!
1 m X X m k ðmÞ r Pi;n k m¼0 k¼0
n ¼ 0; 1; 2; . . . , !
tm ; m!
n ¼ 0; 1; 2; . . . ,
where PðmÞ i;n is the m-step transition probability of the associated random Markov chain. Further, to make the recurrence Eqs. (3)–(5) simpler, the above equation can be rewritten as qn ðtÞ ¼ eð1þrÞt rni
1 X
cðm; n; iÞ
m¼0
tm ; m!
n ¼ 0; 1; 2; . . . .
(6)
It should be noted that qn(t) is completely determined once cðm; n; iÞ is known. Therefore, substituting from Eq. (6) into Eqs. (3)–(5), we get the following recurrence relations: cðm þ 1; 2n; iÞ ¼ r1 cðm; 2n 1; iÞ þ r2 cðm; 2n þ 1; iÞ; cðm þ 1; 2n þ 1; iÞ ¼ cðm; 2n; iÞ þ r cðm; 2n þ 2; iÞ;
n ¼ 0; 1; 2; . . . ,
n ¼ 0; 1; 2; . . . ,
(7) (8)
with cð0; n; iÞ ¼ din .
(9)
After a little algebra, it can be seen that cðm; n; iÞ is a polynomial in r and can be expressed as given in the following theorem. Theorem 2. For any integers n, i, and m, mX0, we get 8 0; m n þ i odd; > < ðmnþiÞ=2 ½ P cðm; n; iÞ ¼ > Bðm; n; rÞr2r ; m n þ i even; : r¼0
where 8 < A m; r; r þ niþ1 r½ðniÞ=2 ; 2 Bðm; n; rÞ ¼ : A m; r þ ni ; r r½ðniþ1Þ=2 ; 2 (i)
m ! Aðm; x; yÞ ¼
i
even;
i
odd:
mþ1 !
2
2
x
y
.
(ii) Að2m þ 1; x; yÞ ¼ Að2m; x; yÞ þ Að2m; x; y 1Þ. (iii) Að2m þ 2; x; yÞ ¼ Að2m þ 1; x; yÞ þ Að2m þ 1; x 1; yÞ. y Py whenever yox, or xo0 or xo0 and yo0. (iv) x . . . ¼ 0 ¼ x (v) [x] stands for the greatest integer not greater than x.
(10)
ARTICLE IN PRESS A.M.K. Tarabia, A.H. El-Baz / Physica A 382 (2007) 430–438
433
Proof. We will carry out a complete induction on m. It can be easily verified that for simple cases m ¼ 0, 1, 2 formula (10) satisfies the recurrence relations (7)–(9). For example if i ¼ 1; n ¼ 0, and m ¼ 1, we have RHS ¼ cð1; 0; 1Þ þ rcð1; 2; 1Þ ¼ r2 þ r r1 ¼ 1 þ r2 ¼ cð2; 1; 1Þ ¼ LHS. Also, for i ¼ 2; n ¼ 1, and m ¼ 1, we get RHS ¼ r1 cð1; 1; 2Þ þ r2 cð1; 3; 2Þ ¼ r1 r þ r2 ð1Þ ¼ 1 þ r2 ¼ cð2; 2; 2Þ ¼ LHS:
&
In general, there are two possibilities viz either m is even or odd. Suppose m is even say 2k; we have the following cases: Case 1: If i is even say 2 l, we have from Eq. (7) RHS ¼ r1 cð2k; 2j 1; 2lÞ þ r2 cð2k; 2j þ 1; 2lÞ. Substituting from (10), we get RHS ¼ r1 :0 þ r2 :0 ¼ 0 ¼ cðm þ 1; 2j; iÞ. Similarly, from Eq. (8), we have RHS ¼ cð2k; 2j; 2lÞ þ rcð2k; 2j þ 2; 2lÞ ¼
kjþl X
Að2k; r; r þ j l Þr2rjþl þ
r¼0
kjþl1 X
Að2k; r; r þ j l þ 1Þr2rjþl1 :
r¼0
Using the fact that A(2k, kj+l, k+1) ¼ 0, which implies that RHS ¼
kjþl X
Að2k; r; r þ j l Þ þ Að2k; r; r þ j l þ 1Þ r2rjþl
r¼0
¼
kjþl X
Að2k þ 1; r; r þ j l þ 1Þr2rjþl
r¼0
¼ cðm þ 1; 2j þ 1; iÞ. Case 2: If i is odd say 2l+1, lX0, we have from Eq. (7) RHS ¼ r1 cð2k; 2j 1; 2l þ 1Þ þ r2 cð2k; 2j þ 1; 2l þ 1Þ ¼ r1
kjþlþ1 X
Að2k; r þ j l 1; rÞr2rjþlþ1 þ r2
r¼0
kjþl X
Að2k; r þ j l; rÞr2rjþl :
r¼0
Replacing r by r1 in the second term, we get RHS ¼
kjþlþ1 X
Að2k; r þ j l 1; rÞ þ Að2k; r þ j l 1; r 1Þ r2rjþl
r¼0
¼
kjþlþ1 X
Að2k þ 1; r þ j l 1; rÞr2rjþl
r¼0
¼ cðm þ 1; 2j; iÞ.
ARTICLE IN PRESS A.M.K. Tarabia, A.H. El-Baz / Physica A 382 (2007) 430–438
434
Using Eq. (8), we get RHS ¼ cð2k; 2j; iÞ þ rcð2k; 2j þ 2; iÞ ¼
kjl1 X
Að2k; r; r þ j þ l þ 1Þr2rjþ3lþ2 þ r
kjl2 X
r¼0
Að2k; r; r þ j þ l þ 2Þr2rjþ3lþ1 .
r¼0
Apply the fact that Að2k; k j l 1; k þ 1Þ ¼ 0 to obtain RHS ¼
kjl1 X
Að2k; r; r þ j þ l þ 1Þ þ Að2k; r; r þ j þ l þ 2Þ r2rjþ3lþ2
r¼0
¼
kjl1 X
Að2k þ 1; r; r þ j þ l þ 2Þr2rjþ3lþ2
r¼0
¼ cðm þ 1; 2j þ 1; iÞ. Working likewise for the case m ¼ 2k+1 (say) and is omitted here for brevity. Hence, result (10) is fully established and thus, the solution of the differential-difference equations (3)–(5) can be written in the following cases: Case 1: If i is even say (2l) q2n ðtÞ ¼ eðlþmÞt
1 mnþl X X
Að2m; r; r þ n l Þr2rþnl
m¼0 r¼0
q2nþ1 ðtÞ ¼ eðlþmÞt
1 mnþl X X
ðmtÞ2m ; ð2mÞ!
n ¼ 0; 1; 2; . . . ,
(11)
Að2m þ 1; r; r þ n l þ 1Þr2rþnlþ1
ðmtÞ2mþ1 ; ð2m þ 1Þ!
n ¼ 0; 1; 2; . . . .
(12)
Að2m þ 1; r þ n l 1; rÞr2rþnl1
ðmtÞ2mþ1 ; ð2m þ 1Þ!
n ¼ 0; 1; 2; . . . ,
(13)
m¼0 r¼0
Case 2: If i is odd say (2l+1) q2n ðtÞ ¼ eðlþmÞt
1 mnþlþ1 X X m¼0
q2nþ1 ðtÞ ¼ eðlþmÞt
r¼0
1 mnþl X X
Að2m; r þ n l; rÞr2rþnl
m¼0 r¼0
ðmtÞ2m ; ð2mÞ!
n ¼ 0; 1; 2; . . . .
(14)
3. Special cases Case (a): Conolly et al. [5] formulae. In the following lemma, we will show that Eqs. (11) and (12) can be reduced to Conolly et al. [5] formulae. Lemma 1. Conolly et al. [5] formulae are a special case of our formulae (11) and (12) when i ¼ 0. Proof. Put i ¼ 0 in Eqs. (11) and (12), we obtain q2n ðtÞ ¼ eðlþmÞt
1 X mn X
Að2m; r; r þ nÞr2rþn
m¼0 r¼0
q2nþ1 ðtÞ ¼ eðlþmÞt
1 X mn X
ðmtÞ2m ; ð2mÞ!
n ¼ 0; 1; 2; . . . ,
Að2m þ 1; r; r þ n þ 1Þr2rþnþ1
m¼0 r¼0
ðmtÞ2mþ1 ; ð2m þ 1Þ!
n ¼ 0; 1; 2; . . . .
(15)
(16)
Replacing r by rn in Eqs. (15) and (16), we get q2n ðtÞ ¼ eðlþmÞt
1 X m X m¼0 r¼n
Að2m; r n; rÞr2rn
ðmtÞ2m , ð2mÞ!
(17)
ARTICLE IN PRESS A.M.K. Tarabia, A.H. El-Baz / Physica A 382 (2007) 430–438
q2nþ1 ðtÞ ¼ eðlþmÞt
1 X m X
Að2m þ 1; r n; r þ 1Þr2rnþ1
m¼0 r¼n
435
ðmtÞ2mþ1 . ð2m þ 1Þ!
(18)
Now, we have m X
Að2m; r n; rÞr
r¼n
m ¼
!
m
¼
! r þ
þ þ
m
r¼n
rn
!
m
mn2
m
m2
! r2rn
r
! r
nþ1 ! m
1 !
m
!
m X
m
n
n
0
2rn
nþ2
m þ
2mn4
r
!
!
m nþ2
2
m þ
mn1
rnþ4 !
!
m
r
m1
mn2
m þ
!
mn
m
!
m
r2mn .
Rearrangement the power of r and using the fact that m m ¼ , x mx we get m ¼
!
0
m n
! 2mn
r
m þ
m þ þ
mn2
!
!
1 m
!
m nþ1 !
m2
r
r
nþ4
2mn2
m þ
!
2
m þ
mn1
!
m
!
nþ2 m m1
r2mn4 ! r
nþ2
m þ
mn
!
m
!
m
rn
¼ Að2m; 0; nÞr2mn þ Að2m; 1; n þ 1Þr2mn2 þ Að2m; 2; n þ 2Þr2mn4 þ þ Að2m; m 2; m n 2Þrnþ4 þ Að2m; m 1; m n 1Þrnþ2 þ Að2m; m; m nÞrn m X ¼ Að2m; r n; rÞr2m2rþn . r¼n
Hence, Eq. (17) can be rewritten as follows: q2n ðtÞ ¼ eðlþmÞt
1 X m X m¼0 r¼n
Að2m; r n; rÞr2m2rþn
ðmtÞ2m . ð2mÞ!
(19)
Substituting by the values of A(2m,rn,r) and A(2m+1,rn,r+1) into Eqs. (19) and (18) respectively, Conolly et al. [5] formulae can be deduced. & Case (b): Symmetric random walk. If r ¼ 1 we will show that Eqs. (11)–(14) can be reduced to the coefficient of the well-known solution of unrestricted symmetric random walk. To achieve this, setting r ¼ 1 in formula (10) yields: Lemma 2. For any integers n, i, and m, mX0, and for r ¼ 1, we get 8 0; m n þ i odd; > > ! < m cðm; n; iÞ ¼ mnþi m n þ i even: > > : 2 Proof. Consider the case m ¼ 2k, n ¼ 2j, and i ¼ 2l, we get ! kjþl ½ðmnþiÞ=2 X X k k niþ1 cðm; n; iÞ ¼ . A m; r; r þ ¼ rþjl 2 r r¼0 r¼0
ARTICLE IN PRESS A.M.K. Tarabia, A.H. El-Baz / Physica A 382 (2007) 430–438
436
Using Riordan [9], we obtain !
2k
cðm; n; iÞ ¼
m
;
kjþl
!
mnþi .
¼
2
In the same manner, the proof of this proposition can be completed for the other cases. Hence, for any i, we have !
m
mnþi .
cðm; n; iÞ ¼
2
On account of the above proposition and if i ¼ 2l, we obtain q2n ðtÞ ¼ e2mt q2nþ1 ðtÞ ¼ e
2mt
!
1 X
2m
m¼0
mnþl
1 X
2m þ 1
m¼0
mnþl
!
ðmtÞ2m ; 2m!
n ¼ 0; 1; 2; . . . ,
ðmtÞ2mþ1 ; ð2m þ 1Þ!
n ¼ 0; 1; 2; . . . .
Similarly, if i ¼ 2l+1, we have q2n ðtÞ ¼ e
2mt
q2nþ1 ðtÞ ¼ e2mt
!
1 X
2m þ 1
m¼0
mnþl
1 X
2m þ 1
m¼0
mnþl
!
ðmtÞ2m ; 2m!
n ¼ 0; = 1; 2; . . . ,
ðmtÞ2mþ1 ; ð2m þ 1Þ!
n ¼ 0; 1; 2; . . . .
If i ¼ 0 and r ¼ 1, we have q0 ðtÞ ¼ e2mt
1 X 2m ðmtÞ2m m¼0
m
2m!
¼ e2mt I 0 ð2mtÞ,
1.2 q_8 Transient probabilities
1
q_6 q_4
0.8
q_2
0.6
q0 q2
0.4
q4 q6
0.2
q8 0 0
50
100
150
200
250
300
350
400
Values of time t Fig. 1. Transient probabilities of some even states when the system starting at i ¼ 2, l ¼ 0.1, m ¼ 0.2.
ARTICLE IN PRESS A.M.K. Tarabia, A.H. El-Baz / Physica A 382 (2007) 430–438
437
P 2m 2 where I 0 ð2mtÞ ¼ 1 m¼0 ððmtÞ =ðm!Þ Þ is a modified Bessel function (see Ref. [10]). Clearly, this agrees with Conolly et al. [5]. &
4. Numerical illustrations In this section, the computational results obtained by employing the above approach are discussed through three graphs. In Figs. 1 and 2 the transient probabilities of the even and odd states, respectively, are plotted for i ¼ 2, l ¼ 0.1, m ¼ 0.2. As can be seen, the transient behavior goes to zero for n-N and the given system has no steady-state values. In Fig. 3, we show the effects of the traffic intensity r on the transient probability q0(t) when the system is starting at i ¼ 2. Fig. 3 shows that for a given r, the transient probability q0(t) decreases as the parameter r decreases. This is understandable since for such values, the arrival rate is decreased as compared to the service rate, thus forcing the system to have more chance to reaches to zero as n-N.
0.3 q_7
Transient probabilities
0.25
q_5 0.2
q_3 q_1
0.15
q1 q3
0.1
q5 0.05 0
q7
0
100
200
300
400
500
Values of time t Fig. 2. Transient probabilities of some odd states when the system starting at i ¼ 2, l ¼ 0.1, m ¼ 0.2.
0.14 0.12
Probability
0.1
roh=0.1
0.08
roh=0.25 roh=0.5
0.06 0.04 0.02 0 0
100
200
300
400
Values of time t Fig. 3. Effects of the traffic intensity r on the transient probability q0(t) when the system starting at state i ¼ 2.
ARTICLE IN PRESS 438
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Acknowledgments The authors are grateful to the Area Editor and the anonymous referees for their useful comments and suggestions, which helped to improve the earlier version of this paper. References [1] O.P. Sharma, D.B. Bunday, Optimization 40 (1997) 79–84. [2] O.P. Sharma, A.M.K. Tarabia, A simple transient analysis of an M/M/1/N queue, Sankhya¯ , Ser. A 62 (2000) 273–281. [3] A.M.K. Tarabia, A new formula for the transient behaviour of a non-empty M/M/1/N queue, Appl. Math. Comput. 132 (2002) 1–10. [4] A.M.K. Tarabia, A.H. El-Baz, Exact transient solutions of non-Markovian queues, Comput. Math. Appl. 52 (2006) 985–996. [5] B.W. Conolly, P.R. Parthasarathy, S. Dharmaraja, Math. Scientist 22 (1997) 83–91. [6] R.J. Glauber, J. Math. Phys. 4 (1963) 495–508. [7] W.H. Stockmayer, et al., Local-jump models for chain dynamics, Pure Appl. Chem. 26 (1971) 555–561. [8] D. Gross, C.M. Harris, Fundamentals of Queueing Theory, second ed, Wiley, New York, 1985. [9] J. Riordan, Combinatorial Identities, Wiley, New York, 1968. [10] M. Abromowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1970. [11] W.J. Anderson, Continuous-time Markov Chains. An Applications-oriented Approach, Springer, New York, 1991.