A stochastic model for highway traffic

Trunspn Res:B Vol 158, No. 4, pp. 285-294. 1981 Printed m Great Bntain

0191.2615,81~040285-lO802.W/0 0 1981 Pergamon Press Ltd.

A STOCHASTIC MODEL FOR HIGHWAY TRAFFIC?

KARMESHU and R. K. PATHRIA

Department of Physics. University of Waterloo, Waterloo, Ontario, Canada

(Received 12 December 1979)

Abstract-A nonlinear model for unidirectional flow of heavy traffic on a two-lane highway is considered. Features such as entrance, exit and lane transfer with time-dependent parameters are incorporated into the model. with the result that a number of previous models employed in the study of traffic flow become special cases of ours. Using the method of system-size expansion. an asymptotic analysis of the problem, including the time evolution of both deterministic and stochastic aspects of the traffic system, is carried out. In addition, a scheme for obtaining the moments of the probability distribution for systems of finite size is developed and a comparison is made with the exact results appropriate to a particular model. The agreement between the two sets of results turns out to be remarkably good.

1. INTRODUCTION

The importance of traffic flow on a two-lane unidirectional highway can be gauged from the fact that the greater part of inter-city traffic takes place on this type of highways. However, in contrast with its importance, the number of theoretical studies in this area is rather limited. The major reason for this seems to be that in dealing with this problem one has to cope with nonlinear stochastic features which arise from the various interac- tions that come into play in a system with heavy traffic flow. In general, it is the nonlinearity of the phenomenon which makes the analysis of the problem intractable. An especially disturbing aspect of the situation is related to the fact that the evolution equation for each moment of the distribution involves higher moments, thus creating a nonclosed hierarchy of equations to be tackled simultaneously.

Some years ago Gani and Srivastava (1969) examined some stochastic models of traffic congestion on a two-lane unidirectional highway. While they carried out a detailed analysis of the time evolution of the linear version of their model, much progress could not be made in the presence of nonlinearity. Later, Schach (1970) investigated the steady state of a closed model dealing with lane transfers in the absence of entrances and exits. Other models featuring lane interchanges have been discussed by Gazis, Herman and Weiss (1962) and by Oliver and Lam (1965). In all cases the time evolution of the stochastic aspects of the problem has continued to defy solution.

In this paper we consider a generalized nonlinear stochastic model of heavy traffic moving in the same direction on a two-lane highway. The traffic system is assumed to be an open one in the sense that, in addition to lane transfers, both entrance and exit of vehicles are allowed. The formulation of the problem is done in terms of the concentrations ni(t) and nz(t) as stochastic variables and the transition probabilities are assumed to have general forms such that most previous models can be treated as special cases of ours. Since the exact solution of the general nonlinear problem is not possible, we look for an approximate, but analytic, solution which may, for its validity, require an asymptotic condition such as n,, nz >> 1 but may nevertheless possess the desired feature of uncovering the dynamics of the system as a function of time. Such a solution can indeed be found if one employs a method based on the concept of “system-size expansion”, which was initiated by van Kampen (1961) and further investigated by Kubo, Matsuo and Kitahara (1973) in connection with the problems of nonequilibrium statistical mechanics. This method, also known as the diffusion approximation, has been quite in vogue in recent years and has been applied to stochastic models in a variety of disciplines; for

tWork supported in part by the Natural Sciences and Engineering Research Council of Canada.

285

286 KAKMESHU and R. K. PATHRIA

details, see van Kampen (1976). However, to the best of our knowledge, this powerful technique has not been applied so far to analyze problems of traffic flow. Since its use in situations of heavy traffic, which are commonly encountered, seems very promising we feel it is high time to introduce this method into this area as well.

In this method one attempts to solve the master equation of the given process by carrying out a systematic expansion in powers of a small parameter which. in a variety 01 situations, turns out to be proportional to 1-‘- 1’2, where .,I” represents the overall size of the system; the reason for this lies in the fact that the relative fluctuations in most statistical variables pertaining to a many-body system are generally of this order of magnitude. The lowest approximation in this scheme yields the “deterministic, or mean evolution, equation” for the time development of the system and its approach to a steady state. The next approximation has the form of a “Fokker-Planck equation” which deter- mines the time development of fluctuations about the deterministic behavior; the various moments of interest can then be readily evaluated. Higher approximations exist but do not play a vital role in most applications. The mathematical justification of this method can be found in the works of Kurtz (1970, 1971) and McNeil and Schach (1973).

The analysis based on the concept of system-size expansion is asymptotic in character and, in principle, applies only to systems of a fairly large size. This does not, however. present any practical difficulties when we are interested in the study of heavy traffic flows. For moderate and low volume traffic, some adjustments become necessary. In Section 5, we have shown how the asymptotic expressions for the various moments of the probability distribution can be manipulated to calculate “finite size effects” on certain important properties of the system. As an illustration, we have derived finite size corrections to the first and second moments in the case of Schach’s model and find that, even for a total of ten vehicles (which can hardly be regarded as a large number), our numerical results agree remarkably well with the ones obtained from the exact steady-state solution of the

master equation.

2. THE MODEL

We consider unidirectional flow on a two-lane highway with the possibility of entrance and exit from either of the lanes. Let si and s2 denote the maximum (acceptable) capacity, and ni(t) I si and nz(t) I s2 the concentration of vehicles, in lane 1 (slow) and lane 2 (fast), respectively. The transfer of vehicles between the two lanes is an inherent feature of the model and plays a vital role in determining the dynamics of the system. The evolution of the traffic system, under the influence of entrances, exits and lane transfers, may be viewed as a birth and death process. To fix ideas, we first recall the traffic flow model of Gani and Srivastava (1969) whose transition probabilities are given

by

Transition Transition probability (in time &)

(n,,n,+n, + l,n,) Xl(S1 - n,)& + cl(&)

1,n,-+n, - l,Q2)

j::L,n,+n,.,l, + 1) plM16t + o(6r)

xJs2 - 122)6t + o(&) (1) (Hi, n2--+ni,n2 - 1) /?2M& + o(6t)

(Hi, n2 - nl - l,n, + 1) LQzn,(s, - M2W + o(St)

@I, n,-+n, + l,n, - 1) Pz,nz(s, - n,w + 4w

We here propose to consider a generalized model of traffic flow from which the Gani- Srivastava model and several other models will emerge as special cases. With this in mind, we note that for an asymptotic analysis of the problem we must identify, on the basis of physical reasoning, some large parameter ,Y- of the system which is directly proportional to the overall size of the system and, hence, may be adopted as a “scaling parameter” for quantities such as ni , n2, s1 and s2. The corresponding large parameters in statistical physics are generally known as “extensive variables” and it is well known

A stochastic model for highway traffic 281

that, in the thermodynamic limit, the most salient features of a many-body system can be expressed solely in terms of a few “intensive variables” relevant to the problem, the extensive variables playing no special role whatsoever; see, for instance, Pathria (1972). In the same manner, while the existence of a large parameter .JV in the system is vital for the application of the method of system-size expansion, its precise choice is not crucial to the solution of the problem, insofar as the final asymptotic results for quantities such as

nl/sl and n,ls,, which are really the ones of direct practical interest to the traffic engin- eer, do not depend explicitly on the choice of JV.

Nevertheless, in the model of Gani and Srivastava, one may identify .,V with either s1 or s2, both of which are supposed to be large numbers and, quite obviously, are proportional to the overall size of the system. Accordingly, one may write

s1 = ..Fa, , s2 = &$,“a2 (2)

where al and a2 are both O(1). In a closed model, such as of Schach (1970), in which the entrance and exit processes are neglected and hence the total number of vehicles N stays constant, this number itself may be indentified with the desired parameter J’.

We now write our generalized model of traffic flow in terms of the transition probabilities

Transition Transition probability (in time at)

(n1,n2+nl + l,n2)

(nl,n2*nl - Ln2)

(nl, n2 - nl, n2 + 1)

@I, n2 - 4, n2 - 1)

(n,,n2+nl - l,n2 + 1 (nl ,n2-nl + l,n, - 1 I (3)

HereA and gi are the entrance and exit functions of the stochastic variable Hi(t) while hij are the transfer functions of both ni and nj; i, j = 1, 2. In addition, we allow for the

possibility that the transition coefficients depend on t explicitly, which may assume a special importance when one deals with situations pertaining to the rush-hour traffic, The precise form of the functionsfi, gi and hij can be chosen so as to suit the situation on hand. For instance, the model of Gani and Srivastava corresponds to the choice

h(5) = 5nmx - 5, gi(5) = s’, hj(5, tl) = t(~lmax - ‘11 (4)

while that of Schach corresponds to setting

ai = Bi = 0, hij(5, V) a 5’ (5)

Moreover, in each case, the transition coefficients ai, fii and pij are supposed to be independent of time.

It seems worthwhile to emphasize that, despite the appearance of ./lr’ in the first four transition probabilities of the generalized model and JV’ in the last two, all transition probabilities are really of the same order of magnitude; otherwise, some of the processes would become inconsequential. This can be ensured by requiring that, while p12 and pzl are O(l), al, PI, a2 and p2 are O(_,tr); see eqn (7), in which ai and pi have been replaced by yi and 6i, where

yi = ai/~, Si = Bi/~ (i = 1, 2). (6)

In passing, we note that generally the maximum (acceptable) capacity of the slow lane is larger than that of the fast lane (because of the difference in speeds) and the vehicles do not enter or exit from the fast lane. Such situations can be realized by setting a, > a2 and y2 = 62 = 0.

288 KARMESHU and R. K. PATHRIA

3. SYSTEM-SIZE EXPANSION OF THE MASTER EQUATION

The master equation describing the time evolution of the probability distribution ~(n, , n2 ; t) for the generalized model (3) is

+ (EF-’ - l)p12(t)~2h12 2,

+ (E-'F - l)p21(t)A'"2h21 ( 2,

where E and F are the difference operators

n2 -1 Jv”

nl -)]P(“‘. n2; t) Jv- (7)

E”f(n,) =f(n, f l), F”f(n2) =f(n2 k 1). (8)

To obtain the desired expansion of the master equation we transform to new variables x,(t) and x2(t) by setting

n,(t) = ,,Vqb,(t) + ~V”~~~x,(t) (9)

n2(t) = N4,(t) + JIr112x2(t)

where [~#~r(t), $2(t)] are supposed to determine the mean evolution, and [xl(t), x2(t)] the stochastic evolution, of the concentrations of vehicles in the two lanes. We are, therefore, interested in studying the probability distribution P(xr , x2 ; t), where

P(x,, x2 ; t) = Np{M&(t) + M1’2Xl(t), N&(t) + N1!2x2(r); t) (10)

At the same time, we write

df (E” - l)f= ~~“.-‘“~+$+$ __. 1 1

(F” - l)f = +Jv-1’2 g + i_Jv-’ g f . . ) etc. 2

> (11)

Substituting (9-11) into (7) and setting t = JV-~~, we obtain an expansion in powers of Jlr- ‘I2 whose leading terms yield the mean evolution equations

Wl - = ?l(~)fl(41) - ~l(~kll(41) - P12(7)h12(41,42) + P21(7)h21(62,41) dr

d& ~ = Y2(~)f2(42) - b2bb2t42) + P12b)h12(+1, $2) - P21W21(42, 41) dr

(12)

while terms of the next order lead to the Fokker-Planck dependent coefficients

ap x=

equation with time-

(13)


where

A 11 = ~bl(T)fl - &(+7, - P12(&2 + P21WzJ

A 12 = +&- P12wh2 + P21(+211

A 21 = &IP12m2 - PZlW211

A 22 = & CY2(T)f2 - 82(492 + P12(&2 - P2l(eh211

B,, = Yl(m-I + b(% + P12(W2 + P21W21

B12 = B21 = -P12(&2 - P21W21

B22 = Y2(4f2 + 62(4g2 + PI2(@12 + P21(#21

(14)

The structure of eqn (13) shows that, asymptotically, the fluctuations in the system correspond to a nonstationary Ornstein-Uhlenbeck process which leads to a bivariate gaussian distribution as its solution.

If the initial values of n, and n2 are well defined, so that P(xi, x2 ; 0) = 6(x,)6(x2), then the first moments of the probability distribution P(xi, x 2 ; t) are identically equal to zero, with the result that

E ‘+ = d,&(r), [ 1

E !$ [ 1

The second moments obey the set of equations

&E(x:) = 2A,,(r)E(x:) + 2A12(3&1~2) + BII(~)

(15)

&E(w2) = A21(W(x:) + [A,,(7) + A22(.r)lW1~2) + A,2WE(x:) + &2(r) t

(16)

&E(d) = ~A~IW(XIX~) + 2A22(W(d) + B22(4 .

The solution of eqns (16) gives the covariance matrix of the probability distribution. It need not be emphasized that, in the asymptotic limit, the entire probability distribution can be characterized by the mean values E(xi) and the covariance matrix E(xixj).

4. PARTICULAR CASES

CASE (i): Open systems In this class of systems, there is no constraint on the total number of vehicles in the

two lanes. The model of Gani and Srivastava belongs to this class and can be obtained by choosing the input, output and transfer functions as given by eqns (4) and assuming the parameters yi, 6i and pij to be independent of time. The mean evolution equations then become

d42 __ = y2(a2 - $2) - 6242 + p1241(a2 - 42) - P2l42(Ul - 41) dr .

(17)


These equations are equivalent to the ones obtained by Gani and Srivastava in the deterministic case, who have solved these equations in the linearized version by imposing the condition pi2 = p2i and have examined some special choices of the parameters yl,z and b1,2. We wish to point out that in the particular case when y1 + a1 = yz + L&, the foregoing equations can be solved even when p12 # pzl. We then obtain

&(r) + 42b) = C&(O) + 42(0)1expi -(rI + &bI + “;: 1 iIn [l - exp: -(ljl + 6,)r)]

(18)

which gives a relaxation time of l/(yi + 6,) for the mean traffic concentration to attain the steady state. Combining eqns (17) and (18), one can obtain solutions for &(r) and 42(r) separately. For r 9 l/(yl + 6,),

(19)

and one obtains

Wl x= -(PI2 - P2M - (Yl + 61) + P2lUl + P12U2 -

(P12 - PZlHY14 + Y2U2)

Yl + 61 1 41

+ YlUl + Pzlal [

YlUl + Y2U2

Yl + 61 1 which is the usual Riccati equation whose solutions are well known.

Gani and Srivastava have suggested a recipe for studying the stochastic evolution of the system but it turns out that for any meaningful evaluation one must resort to a cumbersome numerical analysis. In the very simple case sl = s2 = 2, they have calcu- lated the long-run probabilities which resemble a bell-shaped distribution, in qualitative agreement with the bivariate Gaussian distribution resulting from the present method. It seems worthwhile to mention here that our scheme enables one to obtain explicit expressions for the time evolution of stochastic as well as deterministic aspects of the problem.

Other workers have proposed different choices for the transition probabilities entering into this problem; whereas Gazis et al. have assumed a linear model, Oliver and Lam have considered a nonlinear one with transfer functions of the form

hj(59 VI a t2(b~- vl). (21)

The justification for these forms is largely empirical, though the dependence on t2 is clearly reminiscent of the (a/u2)-term in the van der Waals equation of state of a gaseous system. Munjal and Hsu (1973) have carried out an empirical comparison of the linear model of Gazis et al. and the nonlinear model of Oliver and Lam and have found the nonlinear model to be preferable. A similar conclusion has also been reached by Chang and Gazis (1975). In this context, we feel that the form of the entrance functions 5 may also be modified to

fi(5) K (&%JX - 12) (22)

because, when the traffic density is low, the reluctance of an average driver on entering a highway is not so sensitive to the actual value of 5 as suggested by the choice (4). The precise consequences of these modifications on the results of our analysis are presently under investigation and will be reported in a subsequent communication.

CASE (ii): Closed systems In this class of systems, the total number of vehicles involved in the traffic is conserved,

which is generally a consequence of neglecting the entrance and exit processes. The


model of Schach belongs to this class and can be obtained by choosing yi = 6i = 0 (which implies that nl(t) + nz(t) = N) and h,,{&q) = Ar, where A is a constant. The straightforward implications of the constancy of N are that (i) the problem reduces to a single-variable one in, say, nl(t) and (ii) the large parameter JV can be identified with N itself. The system-size expansion is then effected by writing nl(t) = N@,(t) + N1”xl(r) and going through the conventional steps. Schach himself analyzed his model in the steady state and showed that, as N tends to infinity, the long-run concentration of vehicles in either of the lanes follows a gaussian distribution, in complete agreement with the findings of the present analysis. However, no attempt was made to study the time evolution of the system.

The mean evolution equation in this case is rather straightforward:

(23)

where & = 1 - 41. The steady-state values of the mean numbers of vehicles in the two lanes are given by

E(q) = Nc#I, = $$; E(n,) = N$J~ = +$ (24)

where

a = p;‘t and b = p;‘,i’.

The equation for the second moment is

&E(d) = W11 - A,,)@:) + BII

where

A 11 = - vi24i-‘, AIZ = ~P;I&-', B1i = P;z& + Pii&.

The steady-state value of the variance of nl, as well as of n2, turns out to be

E(Cni - E(ni)l’) = r(fTbb)2 (i = 1,2).

(25)

The probability distribution functions of ni are, therefore, Gaussian with mean and variance given by eqns (24) and (26).

In passing, we note that in the method presented here the time development of the various quantities of interest pertaining to the system can be evaluated in a straightforward manner.

5. FINITE SIZE CORRECTIONS TO MOMENTS

Strictly speaking, the method of system-size expansion is valid only in the asymptotic limit JV -+ 00. In practice, however, the results obtained through this method can be applied with considerable success to situations where JV is finite, so long as it is fairly large. We find that, in the case of a nonlinear model, some of the results can be improved remarkably by manipulating the asymptotic expressions of the various moments, so that the resulting expressions give useful results even when J-is not so large.

An important feature of a nonlinear model lies in the existence of a coupled hierarchy of moment equations. We shall show how this hierarchy may be truncated so as to yield finite-size corrections to the hitherto asymptotic results. As an illustration, we consider Schach’s model with r = 2. The master equation now takes the form

d(n 1 ; t) = (E - l)p’t,~:~(nt ; t) + (E-’ - ~)P;IW - nd2~h ; t). (27)


This leads to the following hierarchy of rate equations for the various moments of the probability distribution p(n, ; t):

$(n:) = 2(Pil - AJE(G) + (A2 + Pii - 4p;1N)E(G) + 2Pi,N(N - l)E(n1)

+ P;IN~ (29)

iEM) = 3(~;1 - p’rAE(n?) + 3(PL + piI - 2pi,N)E(nf) + (PiI - pi2 - 6p;,N

+ 3pilN2)E(nf) + pi1N(3N - 2)Eh) + piIN (30)

etc. It will be noted that the rate equation for E(nl) involves E(nf), that for E(nf) involves E(n:), and so on. This means that to obtain E(nlj one has to know E(n:), to obtain E(nf) one has to know E(n:), and so on. To obtain exact moments from such an hierarchy is clearly impossible. Considerable progress can be made, however, if one adopts the truncation procedure suggested below. For simplicity, we consider the steady-state moments alone.

Using the relation

n1 = NL a+b

+ N”2~ 1 (31)

where a = pi’/‘, b = p;‘i’ and N = n, + n2, eqns (28)-(30) yield the following exact relations among the various moments:

(32)

which shows that for a system of finite size the expected value of x1 is O(N - ‘j2),

(a’ - b2)N”2E(x;) = ab(2N - l)E(x:) - & N (33)

(a2 - b2)2E(x:) = 4a2b2N - ab(a2 + 4ab + b2) + ab(a2 + b2) i 1 E(xf)

2a3b3 N _ a2b2(a2 + b2) - (a (a + b)2 1

etc. In the asymptotic limit, eqn (34) gives

(34)

E(x:) = ab

2(a + b)2 ’

in agreement with (26). Using this value of E(x:) in eqn (32), we obtain, in the first approximation

-WI) 1 ,;,“,, N-'/2. (35)

For a better approximation, we replace E(x:) in eqn (34) by its asymptotic value, viz. 3 [ab/2(a + b)2]2, with the result

E(xf) 2: ab 1

2(a + b)’ + 16N (36)

A stochastic model for highway traffic

This not only improves (35) to

293

but also yields, with the help of (33),

E(x:) rz a&J - b) N-‘,z

8(a + b)3 ’ (37)

It will be noted that, for large N,

E(x;) N E(x,) E(xf). (38)

From (35a) and (36), we obtain for the mean and the variance of nl

u-b 1 a’-b2 1

32abNZ

1 - var(n,) ‘v

ab N 2(a + b)’

(39)

It is quite understandable that the improvement in the value of E(nl) is one order of magnitude higher than that in var(n,). The foregoing results may be compared with similar ones obtained by van Kampen (1976) for a model of a semiconductor by examining higher-order corrections involved.

At this point we observe that, approximation,

+m = 3,

to the Fokker-Planck equation for the processes

in the special case a = b, we obtain, without

kvar(n,) = 8 1 - & ( >

-1

. (41)

Table 1 shows a comparison of the numerical values of N-‘E(n,) and N-‘var(n,) obtained by using our approximate expressions (39) and (40) with the ones obtained by using the exact probability distribution derived by Schach. Agreement between the two sets of results is remarkably good, even for N = 10 which can hardly be regarded as a large number.

6. DISCUSSION

We have applied the method of system-size expansion for carrying out a systematic analysis of stochastic processes occurring in a traffic system of large size. The chief merit

Table 1. Numerical values of fraction f[ = N- ‘E(N~)] and variance u[ = N- ‘var(n,)] for Schach’s model with r = 2t

a/b F 2

1 3

T -

I

I

1

+since the inrerchange Of a and b produces cmlp1ementary results, we have tabulated values for (a/b) > 1

294 KAKMESHU and R. K. PATHRIA

of this method lies in the fact that it enables us to obtain a complete picture of the time evolution of the stochastic as well as deterministic aspects of the problem-under the sole restriction that the number of vehicles involved is large enough to justify the expansion procedure. This is hardly a serious restriction in heavy-traffic situations. Evaluation of finite-size corrections leads to a remarkable improvement in the theoretical results, which is illustrated by a comparison with the exact results pertaining to Schach’s model. It is obvious that our analysis applies equally well to situations where the parameters of the problem are explicitly time-dependent. Such situations are encountered in particular when one deals with the rush-hour phenomena. A detailed study concerning this aspect of the problem is in progress.

The model discussed in this paper can be readily generalized to three-lane, or even multi-lane, traffic and analyzed using techniques employed here. An interesting aspect of

such studies may be the existence of limit cycles in traffic flow. We are currently looking into this aspect as well.

REFERENCES

Chang M. Fand Gazis D. C. (1975) Traffic density estimation with consideration of lane changing. Trunspu Sci. 9, 308-320.

Gani J. and Srivastava R. C. (1969) Some models for traffic congestion. Tra~tspn Res. 3, 333-344. Gazis D. C.. Herman R. and Weiss G. H. (1962) Density oscillations between lanes of a multilane highway.

Opns Rex 10, 658-667. Kubo R., Matsuo K. and Kitahara K. (1973) Fluctuation and relaxation of macrovariables. J. Stat. Phys. 9,

51-96. Kurtz T. G. (1970) Solutions of ordinary differential equations as limits of pure jump Markov processes. J.

Appl. Proh. 7, 49-58. Kurtz T. G. (1971) Limit theorems for sequences of jump Markov processes approximating ordinary differential

processes. J. Appl. Proh. 8, 344-356. McNeil D. R. and Schach S. (1973) Central limit analogues for Markov population processes. J.R. Stutist. Sot.

B35, 1-23. Munjal P. K. and Hsu Y. S. (1973) Experimental data validation of lane-changing hypotheses from aerial data.

Highway Research Rec. No. 456, pp. 8-19. Oliver R. M. and Lam T. (1965). Statistical experiments with a two-lane flow model. in the Proc. 3rd Int. Symp.

Theory of Traf%c Flow, pp. 17&180. Elsevier, New York. Pathria R. K. (1972) Statisticul Mechanics Pergamon Press, Oxford. Schach S. (1970) Markov models for multi-lane freeway traffic. Transpn Res. 4, 259-266. van Kampen N. G. (1961) A power series expansion of the master equation. Can. J. Phys. 39, 551~567. van Kampen N. G. (1976) The expansion of the master equation. Adc. Chem. Phys. 34,245-309.

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A stochastic model for highway traffic