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1 Queues with Time-Dependent Arrival Rates I:The Transition through Saturation Author(s): G. F. Newell Source: Journal of Applied Probability, Vol. 5, No. 2 (Aug., 1968), pp Published by: Applied Probability Trust Stable URL: Accessed: :33 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Applied Probability Trust is collaborating with JSTOR to digitize, preserve and extend access to Journal of Applied Probability

2 J. Appl. Prob. 5, (1968) Printed in Israel QUEUES WITH TIME-DEPENDENT ARRIVAL RATES I-THE TRANSITION THROUGH SATURATION G. F. NEWELL, Institute of Transportation and Traffic Engineering, University of California, Berkeley Abstract Suppose that the arrival rate A(t) of customers to a service facility increases with time at a nearly constant rate, da(t)/dt = a, so as to pass through the saturation condition,,a(t)= =t = service capacity, at some time which we label as t = 0. The stochastic properties of the queue are investigated here through use of the diffusion approximation (Fokker-Planck equation). It is shown that there is a characteristic time T proportional to a-2/3 such that if t < 0, jt T, then the queue distribution stays close to the prevailing equilibrium distribution associated with the A(t) and #,, evaluated at time t. For it = O(T), however, the mean queue length is much less than the equilibrium mean, and is measured in units of some characteristic length L which is proportional to a-1/3. For t > 0, It 1> T, the queue is approximately normally distributed with a mean of the order L larger than that predicted by deterministic queueing models. Numerical estimates are given for the mean and variance of the distribution for all t. The queue distributions are also evaluated in non-dimensional units. 1. Introduction It is quite common in practical queueing situations that customers arrive at a service facility with an arrival rate A(t) which is significantly time dependent. Frequently the arrival rate gradually increases as a "rush hour" approaches until 2(t) actually exceeds the service capacity of the server. Eventually the arrival rate subsides again and the queue which builds up during the rush hour is worked off. Situations of this type are customarily analyzed only by means of deterministic queueing models [1-3]. We draw a graph of the expected cumulative number of arrivals to time t, E{A(t)}, versus t as in Figure 1 and treat this as if it were the actual number of arrivals. As long as the slope of this curve, the arrival rate 2(t), stays less than the service rate p, we approximate the queue to be zero and say that the (expected) cumulative number of departures to time t, E{D(t)}, is equal to E{A(t)}. But as soon as A(t) exceeds p, we draw the expected departure curve with a slope p (also treated as if it were the actual departure curve) until E{D(t)} is again equal to E{A(t)}, after the rush hour is over. Thus the (possibly Received'21 October

3 Queues with time-dependent arrival rates. I: The transition through saturation 437 / 3 i SI / S Arrivals // I Departure E Slope=X(f) / Slope=~ 0 t, Time -t Figure 1 stochastic) queueing phenomena are represented as if customers were a fluid which is poured into a reservoir having a capacity flow rate p. The expected queue length at any time t (fluid in the reservoir) is E{A(t) - D(t)}, the vertical distance between the two curves. To analyze some of the consequences of statistical fluctuations in the arrival and service rates for a situation like this, we should interpret A(t) and D(t) as random functions of time, the expectations of which have time derivatives A(t) and ~ respectively; the latter, however, only when the actual queue is nonzero. If we define t = 0 as the time when A(t) = y, we would not anticipate a vanishing expected queue even for t <0, certainly not for t10o. It is also clear that the expected queue length at t = 0 must depend upon the rate at which the arrival rate builds up to the value u at t = 0. If at one extreme, the arrival rate builds up very suddenly (da(t)/dt = oo) from a value A(t) = 0 for t < 0 to a value of y at t = 0, the queue will be zero as t t0. On the other hand, if the arrival rate increases very gradually (da(t)/dt -+~0) the queue length distribution will try to adjust to the equilibrium queue distribution at the prevailing traffic intensity, )(t)/p. The equilibrium mean queue length becomes infinite, however, as the traffic intensity approaches 1 at t = 0. For any finite rate of approach to the critical arrival rate, the expected queue length at t = 0 must have some finite value between these two extremes of 0 and oo. Deterministic queueing approximations always tend to underestimate the expected queue length. In particular for t > 0, the expected departure curve would have slope exactly y if the departure process could proceed independently of the queue length, even if the queue length wanted to become negative. For the real queueing process, however, the service might be interrupted if the queue

4 438 G. F. NEWELL vanishes. The actual average service rate can never be larger than Yu. During the time when the deterministic model predicts a nonzero queue, the difference between the actual expected queue length and the deterministic estimate is therefore nondecreasing in time, and is at least as large for t > 0 as the actual expected queue at t = 0, i.e., somewhere between 0 and oo. Even some crude estimates of the order of magnitude of this difference would obviously be of some practical value. 2. Diffusion approximation There is little likelihood that anyone will find a queueing model with a time dependent arrival rate of the above type which is sufficiently simple as to yield exact solutions in any usable form. We shall, therefore, immediately attack this problem by approximate methods. The "second order" approximation, after the deterministic model, is the "diffusion" model [4], [5]. This is also based upon the postulate that the queues are almost always long enough so that one can disregard the discrete nature of customers. In fact, the queues are so long that one can find a scale of time z sufficiently short that during a time z the fractional change in the queue during time z is small (most of the time), yet z is sufficiently large that during time z there are many arrivals and departures from the queue. Furthermore, the number of arrivals and departures during z are approximately normally distributed. The corresponding postulates for the deterministic approximation would imply that r was long enough that one could apply a law of large numbers instead of a central limit theorem, thus neglecting fluctuations completely. These approximations obviously are not accurate when the traffic intensity is well below 1 for t < 0, but the queue is too small to be of any interest then anyway. These postulates lead to a diffusion equation (Fokker-Planck equation) for the queue length. If we assume that behaviors of the arrival and service do not depend explicitly on the queue length (except when the queue length is zero), we let and X(t) = queue length at time t, (1) F(x, t) = P(X(t) < x} be the distribution function for the queue length at time t, then F(x, t) satisfies (approximately) an equation of the type, [6], ~(2) F(x, t) = aof(x, t) b(t) 02F(x, t) at ax + 2 ax2 This equation differs from the more familiar differential equation for a homogeneous process only in that the coefficients a(t) and b(t) depend upon t here.

5 Queues with time-dependent arrival rates. I: The transition through saturation 439 In terms of the arrival and departure processes, the coefficients a(t) and b(t) are given by (3) a(t) = z-'1e{d(t + T) - D(t) - A(t + z) + A(t)}, b(t) = z-1var {D(t + z) - D(t) - A(t + z) + A(t)}, i.e., they are respectively the rate of change in the expected queue length and rate of change of the variance of the queue length. It is implied in (3) that there exists a certain range of r over which the right hand sides of (3) are essentially independent of z. This z is small compared with times over which the a(t) and b(t) change significantly, so in this sense a(t) and b(t) are instantaneous rates on the time scale of t in (2). However, z is large enough that many customers arrive or depart during z, so as to give the asymptotic normal approximation and a variance approximately proportional to z in (3). (The deterministic approximation corresponds to neglect of the second derivative term in (2)). In terms of the 1(t) and i (4) a(t) = p - A(t). For the most common types of stochastic arrival and departure processes, the variances and means of cumulative counts for arrivals or departures are comparable. For a Poisson process they are exactly equal. It is convenient then to write (5) b(t) = I,;(t) + I, for some suitable coefficients I, and I, which we would expect to be comparable with 1 and essentially independent of the 2(t) or p. Equation (2) is to be solved subject to the boundary condition that (6a) F(x,t)- 1 for x-*+co, (6b) F(x,t)-+O for x-+0, and (6c) F(x,t)-+l for x>0, t-+ -oo. Condition (6a) is the necessary condition for a proper random variable. Condition (6b) implies that X(t)> O0. Actually one could admit a slight discontinuity in F(x, t) at x = 0, but to justify the approximations used here, the queue should almost always be large and have very small probability of being exactly zero. Condition (6c) implies that at some time in the distant past when the traffic intensity was low, the queue length was negligibly small. It is still quite difficult to solve even these approximate equations (2) and (6),

6 440 G. F. NEWELL but we can use these equations along with some logical interpretations of queueing phenomena to gain further understanding of what is taking place. 3. Undersaturated conditions Consider first the queue behavior for t < 0, when a(t) > 0. Let us suppose that there is a period of time during which a(t) and b(t) are nearly constant with values ao and bo, respectively. The queue distribution will, given sufficient time, approach an equilibrium distribution obtained by setting OF/Ot = 0 in (2), namely (7) Fo(x) = 1 - exp[-2aox/bo]. This is the continuum exponential approximation to the geometric discrete distribution and has a mean (8) E{X0 = + -oip bo IPo + +I0 I2o0 2ao- 2,uo[1-0o/o] It is, of course, a formula of this type that tells us that for 0/p0o -- 1, the equilibrium queue length becomes infinite. Equally important here, however, is the rate of approach to the equilibrium. Detailed properties of the transient behavior have been considered by Gaver [8]. It suffices here to notice that if we choose new units of time and queue length (9) t' = t/to, x' = x/lo, then (2) becomes OF Toao OF Tobo 02F (10) at' -Lo ax' 2L? Ox'2 If, in particular, we choose To and Lo so that i.e., Toao - 1 and Tob 1, Lo Lo (11) Lo = bo and To = ao ao then the coefficients in (10) become 1 and 1/2. If in this nondimensional form of the diffusion equation we take as an initial state any queue length of order x' = 1, or any nonequilibrium distribution of queue lengths over a range x' of order 1, this distribution will relax to the equilibrium distribution within a time t' of order 1. This must be so since there are no parameters in the equation other than constants of order 1. In terms of the original units, we see that Lo is comparable with (actually two times) the average

7 Queues with time-dependent arrival rates. I: The transition through saturation 441 equilibrium queue length, as would be expected. The unit of time, however, is proportional to ao 2. As the traffic intensity approaches 1, ao -+0, the relaxation time becomes large like ao2 whereas the queue length becomes large only as -1 a0 Let us now return to the question of what happens if a(t) and b(t) (particularly a(t)) are time dependent, slowly varying (in some sense) functions of time with a(t) -+ 0 as t If at some time to we have coefficients a(to) = ao, b(to) = bo, and we change units of time (relative to to as a new origin) and queue length then (2) becomes t = to + Tot', x = x'lo, af a(to + Tot') af b(to + Tot') a2f at' ao 0 x' 2bo x'92 If the coefficients in this equation, considered as functions of t', are nearly constant over a t' range of order 1, for example if and 1 da(to + Tot') I TO da(t)?1 ao dt' ao dt 1 db(to + Tot') I TO db(t)i bo dt' bo dt ' then any deviations from an equilibrium distribution will have time to relax before the equilibrium distribution can change very much. We thus conclude that for any range of time where (12) b(t) da(t) 1, 1 db(t) I<< and a(t)>o, a3(t) dt Ila2(t) dt the queue length distribution will at all times stay close to the prevailing equilibrium distribution, i.e., (13) F(x, t) ~ 1I - exp [- 2a(t)x/b(t)]. In practical applications there is no reason why as t -+0, A(t) should increase in any way other than linearly. It is reasonable to postulate, therefore, that in some neighborhood of t = 0 (14) a(t) = - (t) ~ - at for some constant t, and b(t) = I M(t) + I,g z (IA + I,)M + IRclt.

8 442 G. F. NEWELL In this case conditions (12) for the validity of (13) are that (5at (I + l)a) 1 Ot t Ift)1/2 (15)? 22 and t < 0. The quantity a/#2 in (15) is a dimensionless parameter which can be interpreted as the fractional change in arrival rate during an interarrival time. The diffusion equation is meaningless unless this is very small compared with 1. In all practical cases or any meaningful application of the approximations the second inequality above is implied by the first, i.e., the time dependence of the b(t) is of little consequence near t = 0. The implication of the above conditions are that as t approaches zero and the equilibrium average queue length becomes large, there comes a time when the queue can no longer grow fast enough to keep up with the equilibrium distribution. There is a characteristic time T such that as t/t becomes of order 1, (15) breaks down. We will further confirm later that this time T given by at.(1l + I2) 1}2 (16a) T -(I + II)"3 (t22/3 is the "natural" unit of time for describing the queue behavior near t = 0. For t ~ - T, we also expect the queue length to be of the order of magnitude (16b) L - T) =(I+,)2/3 a(- T) a which will become the natural unit of queue length near t = 0. The most interesting feature of these parameters, T and L, are their dependence upon a, the rate of increase of the arrival rate at t = 0. Tis proportional to a- 2/3 and L is proportional to a-1/3. The latter is at least consistent with our earlier prediction that if a = d2(t)fdt were infinite, the queue length would be zero at t = 0; but if a were zero, the mean queue length would be infinite. Why the queue length should involve the (- 1/3) power of a is certainly less obvious. 4. Oversaturated condition Equation (2) was derived, in essence, from a postulate that queue changes are generated from the addition of independent normally distributed random variables, namely arrivals and departures during periods of time of the order of z. The motivation for the introduction of a differential equation to describe this was to give a convenient representation of the boundary condition that X(t) 0

9 Queues with time-dependent arrival rates. I. The transition through saturation 443 for all t. If it were not for this condition prohibiting negative queue lengths, we could very easily have accumulated sums of normal random variables and found the distribution of X(t) directly. For t > 0, the average queue length is certain to increase rapidly with time, and as this happens it also becomes less likely that a fluctuation can cause the queue to vanish. For a range of t values between some as yet unspecified time to until the end of the rush hour (as estimated say by the time at which the queue vanishes according to the deterministic model), there should be a negligible probability that the queue will vanish at any time. The boundary condition X(t)? 0 will not be violated anyway and is redundant. If at time to, we have a queue length xo, then the queue length at time t > to will have a conditional distribution F(x, tl xo, to) given by where df(x, t xo, to) 1 [-( - x- m(t, to))2 (17)=exp dx (2rn)/2a(t,to) [ 2a2(t, to) j (18) m(t, to) = - a(u)du, t (19) a2(t, to) = b(u)du, i.e., it must be normal with the cumulative mean and variances implied by (3). One can readily check that (17) is indeed a solution of (2) for any xo, to.,it also follows that X(t) can be written as the sum of X(to) and a random variable X(t Ito) with distribution F(x, t 0, to),. For sufficiently large t, X(t) will also be approximately normal with (20) E{X(t)} -j A - a(u)du, Var {X(t)} B + b(u)du, for some suitable numbers A and B which are independent of t. We would also expect that A-E {X(O)}, B Var {X(0)}, since the presence of the reflecting barrier at X(t)= 0 for t > 0 is expected to increase the mean and decrease the variance as compared with a free diffusion from the queue X(0) at time 0. If the arrival rate increases linearly with time as in (14), then

10 444 G. F. NEWELL (21) a'(t, 0) =f b(u)du ~ (I + I,)pt, and (22) m(t, 0) = - a(u)du = t2/2. The a(t,o) has the interpretation of being the spread of the distribution from diffusion. It increases as V/t whereas the mean increases as t2. For sufficiently small t, the first dominates, but for larger t the latter dominates. The spread and the mean are comparable when or (I, + I,)1pt ~ 0 2t4, (Ia +I U)1/3 2)2/3 i.e., this occurs when t is of order + T, at which time both a(t, 0) and m(t, 0) are of order L. This must also be the order of magnitude of the time to in (17) because once the mean of the queue length begins to dominate the standard deviation it becomes very unlikely for the queue to vanish. The time T is thus the order of the time when the queue distribution starts to approach a normal distribution and drift away from the boundary at x = Transition behavior If over a range of time It = O(T), the arrival rate can be approximated by a linear function of t as in (14), we can estimate the queue distribution for t < 0 and t I > O(T) from (13). Also, if we know the queue distribution at some time to comparable with + T, we can estimate the queue distribution for any t > to from (17), at least until the arrival rate comes down and the queue gets close to zero again. To find the distribution at time to, however, we must analyze the queue behavior over the transition region ItI = O(T). Over this range of t, we can approximate (2) by (23) OaF af b(o) 02F at +ax 2 Dax2 If we change units of time and length through (24) t* = tit and x* = x/l, this equation reduces to the nondimensional form (25) F - af 1 a2f (25)= -ax* 2 ax*,- which is still subject to the boundary conditions (6).

11 Queues with time-dependent arrival rates. I: The transition through saturation 445 Since (25) contains no parameters other than constants of order 1, we would expect all features of the solution to be measured on a scale of order 1 in x* and t*. This further confirms that the queue lengths in the original units must be of order L over the range t = O(T). From our previous analysis we expect the solution of (25) to behave asymptotically like (26) F(x,t) = 1 - exp[+2t*x*] for t* <0, It*I > 1, but for t* > 0, t* > 1 to be approximately a normal distribution with mean t*2/2 + 0(1) and variance t* + 0(1). There does not seem to be any simple analytic solution of (25), but since the equation contains no variable parameters we need determine only one dimensionless solution which then, by appropriate change of units, can be mapped into a solution for any parameters T, L, etc. Or conversely, if we determine by simulation or any other means, the evolution of any distribution which satisfies an equation like (23), we can map it into a solution of (25) or of any other queueing problem of type (23). 6. Numerical solution The usual method to integrate numerically an equation like (23) or (25) is to replace it by a finite difference equation. This method, however, can also be described in terms of a hypothetical queueing process in discrete time, or, in the more common terminology, of a random walk on a lattice with a reflecting barrier. Suppose we have a sequence of integer valued random variables Xj, j =?.-, -2, -1, 0, + 1, + 2,..., which represents the queue at time j or the position of a random walk at time j. Let XXji + 1 with probability pj Xj+1 = Imax (X - 1,0) with probability 1 - pj. i.e., the Xi describe a random walk which at each stage moves up one step or down one step (if Xj # 0). This differs from the usual random walk with reflecting barrier in that the pj depend upon j. In particular, we will take (27) pj = (1 + aj)/2, (1 - p) = (1 - aj)/2., for some constant a. The distribution functions for the Xj, F(k) = P{X, = k)_ evaluated only at the integer steps, satisfy the finite difference equations

12 446 G. F. NEWELL (28) Fj+ I(k) = pjf,(k - 1) + (1 - pj)fj(k + 1) k - 1, Fj+1(0) = (1 - pi)fj(l). For pj as in (27), and acj < 1, Equation (28) is the discrete analogue of the differential equation af,(k) = ( afj(k) 1 a2fj(k) (28a) c j + 8j ak 2 ak2 Solutions of (27), (28) were evaluated numerically by a computer for several values of a; a = 10-1, 2 x 10-2, and 2 x This serves two purposes. First since (27), (28) can be interpreted as describing the actual evolution of the distribution functions of a real queue, the behavior of this solution as a function of a, as a decreases, gives an indication of the rate of convergence of the exact solution to a limit solution, thus an indication of the range of validity of the arguments described in the previous sections. Secondly, it gives an estimate of the limit solution which should satisfy (28a), and which, by suitable change in scale, can be mapped into a solution of (25). If, in (28), we were to freeze the p, at a constant value p, for all j > n, the solution of (28) would converge to the equilibrium solution (29) F,,0(k) = 1 - [pn(1 - p,)]k+i = 1 - [(1 + an)/(l - an)]'+' provided n < 0. For Inl > T = 1/a2/3, the solution of (28) at time j = n + 1 should already be close to the equilibrium solution (29) for time n. To obtain a numerical solution of (28), we therefore started from an initial distribution at some j < 0 and moderately large value of I j IT, taking as the initial distribution, the distribution Fj_-,0(k). Values of the Fj+ 1(k) were then evaluated iteratively from (28) until j reached some positive value of order T and Fj(O) became negligible, indicating that the reflecting barrier was no longer of any importance. The pi in (28) are defined as the transition probabilities from stage j to j + 1 and we have chosen the pj in (27) so that Po = 1/2. In order to compare the calculations for different a, it seemed appropriate to identify a time t = - 1/2 with the stage j = 0 so that Po = 1/2 was identified with a transition from time - 1/2 to + 1/2. The time t was then rescaled to and k was rescaled to t* = tt = ta2/3= (j - 1/2)a2/3, * = kil = ka1/3 Some of the results of the numerical calculations are shown in Figures 2, 3, and 4. Figure 2 shows the evolution of the mean queue length, Figure 3 the variance, and Figure 4 the distribution functions.

13 Queues with time-dependent arrival rates. 1: The transition through saturation 447 I 13-- I I o 2 -- I I I 2 a=2xl a2 2x Queue- 2 /a=2xio, Time / 7" Figure 2 In Figure 2, the solid line curves shows the mean queue length in units of L versus t* for a = 10-1, 2 x 10-2, and 2 x For o= 10-1, pj varies all the way from 0 to 1 while t* ranges from about -2 to + 2. The mean queue in original integer units at time 0 is about 1. Despite the fact that the assumptions of the previous sections (queues large compared with 1, a nearly constant variance per step over t* = 0(1), etc.) are not true for a = 10-1, the curve for a = 10-1 is surprisingly close to those for a = 2 x 10-2 and 2 x Also shown in Figure 2 is the curve 1/(2t*) which is the mean equilibrium queue length corresponding to (7) or (26) with b(t) = 1 (independent of t) and x(t) = xt. The solid curves should, for -+ 0, be asymptotic to (1/2t*) for t* -+ - oo. The discrepancy between the solid curves and 1/(2t*) for t* ~ -2 is not so much an indication that the queue distribution differs appreciably from its equilibrium distribution; rather it is a reflection of the fact that for a > 0 the effective b(t) is not sufficiently constant over -2 < t* < + 2. The difference between the actual mean and the mean of the correct equilibrium distribution for oa > 0 given by (29) is so small even for t* ~ -1.5 as to hardly show on a graph to the scale of Figure 2. Figure 2, also shows for t > 0 the curve t*2/2 which is the queue length as given by the deterministic queueing theory. The broken line labeled "Queue -t*2/2" is the difference between the queue length for xt = 2 x 10-3 and the curve t*2/2.

14 448 G. F. NEWELL 3 I I 1- I a, a = VII SMean ' C oo 0 I,-? x 't Figure 4 This difference rapidly approaches a constant value of about 0.95 as t* increases. Even for a = 10-1 the difference between the actual mean and t*2/2 is about We thus conclude that the constant A in (20) is (30) A (0.95)L,

15 Queues with time-dependent arrival rates. I: The transition through saturation 449 and (31) E{X(0)} ~ (0.65)L. Figure 3 shows similar curves for the variances of the queue length (in units of 13) versus t*; also the curve 1/(2t*)2 for the variance of the equilibrium distribution with a - 0, and the curve t* for the variance of a diffusion without reflection from x = 0, t = 0 with b(t) = 1. For the discrete model, the variance per step is 4p(1 - pj), and for xc = 10-1, -2 < t* < 2, this variance runs from 0 to 1 and back to 0. The effect of this shows very clearly in the shape of the curve for a = 10-' near t* = + 2. It also shows to a lesser extent in the curve for a = 2 x 10-2which has a shorter range for pj. The curve for a = 2 x 10-3 does have a nearly constant pj over this range of t* and shows an approach to slope 1 as t* increases. Our main concern is with the behavior of the curves where b(t) is nearly constant. At t* = 0, the three curves give very similar values, the curve for a = 2 x 10-3 giving a variance of about The dependence upon a seems to show the consequences of two opposing effects. As a decreases, the variance per step increases for each fixed t* toward the value 1. This tends to increase the variance of the queue length also as a -+0. But as a decreases, so also do the time intervals between steps on the t* scale of time. This diffusion therefore feels the reflecting barrier more often. Reflections off the barrier tend to decrease the variance of the queue length. (These two effects both tend to increase the mean queue length as a decreases.) One consequence of this competition is that the curves of Figure 3 do not approach a limit monotonically as a decreases. Finally, the most important conclusion from Figure 3 is that the variance of the queue length for small a and large t* is approaching a line of slope 1, which is about 0.3 below that of a free diffusion from x = 0, t* = 0. Thus the constant B in (20) is (32) B ~ -(0.3)L2, which is indeed less than Var {X(0)}, (33) Var {X(0)} + (0.32)L2. The reflecting barrier which for t* < 0 could be considered to have caused the queue to exist at t* = 0, will for t* > 0 not only wipe out the variance of the queue which existed at t* = 0 but will, in addition, wipe out part of the variance which the arrival and departure processes are trying to generate for t > 0. Figure 4 shows the distribution functions for the queue length at various values of t* with a = 2 x 10-. Although these distribution functions are, strictly speaking, step functions, they have been smoothed to suggest the shape of the distribution for cc - 0. The scale of queue length is shown in both the original

16 450 G. F. NEWELL integer units and the rescaled units x*. The curves are labeled with both the rescaled time t* and the values of pj. As t* increases from t* ~ to t* 1.92, the distribution changes from one which is nearly geometric (exponential) to one which is nearly normal. The location of the mean for each distribution is represented by a vertical line segment. For negative t*, the mean queue length is considerably larger than the median, but as t* becomes large, the distribution becomes less skewed. 7. Conclusion This paper represents only the first step in an attempt to bridge the gap between the theory of equilibrium queues with stationary stochastic arrivals, and the deterministic theory of queues with time dependent arrivals. We have considered here only one special problem, the transition behavior of the queue as the arrival rate gradually increases and exceeds the service rate. During the transition, which takes a time of order T, (16a), the arrival rate is assumed to increase linearly with time at a rate dl(t)/dt = a. The queue length during the transition is of order L, (16b), which is proportional to a_'/3 As long as a sizeable queue persists after this transition period, the expected queue length will be larger than that predicted by the deterministic theory by about (0.95)L. The queue length will be approximately normally distributed with a variance about (0.3)L2 less than the cumulative variance of the arrival and departure processes from the time the system passed through saturation. From the point of view of practical applications and designs of service systems, one is usually interested in the largest queue one is likely to see during the rush hour. This will occur some time after the transition, in fact at about the time when the arrival rate has decreased so as again to equal the service rate. By this time the deterministic theory predicts a queue length large compared with L(if d2(t)/dt remains nearly constant for at least a time of order T near the saturation time), and therefore represents a reasonable first approximation; the above contributions of order L representing small corrections. Actually, the second most important property of the queue evolution is the fluctuation in the vicinity of the time when the mean has its maximum. This is measured by the standard deviation in the queue length at this time, and this is also large compared with L. By finding corrections to the deterministic theory, we have, in effect, established sufficient conditions for the validity of the deterministic theory. But we also obtain some suggestions as to conditions when it might not be valid. 8. Acknowledgments Some of the numerical calculations for a = 10-1 were done by Philip Rogers. The calculations for a = 2 x 10-2 and a = 2 x 10-3 were programmed for an IBM 1620 computer by Frederick L. Collins.

17 Queues with time-dependent arrival rates. I: The transition through saturation 451 References [1] OLIVER, R. M. AND SAMUEL, A. H. (1962) Reducing letter delays in post offices. Operat. Res. 10, [2] GAZIS, D. C. AND PoTTs, R. B. (1963) The oversaturated intersection. 2nd International Symposium on the Theory of Traffic Flow, London, OECD Paris 1965, [3] MAY, A. D. AND KELLER, H. E. M. (1967) A deterministic queueing model. Transportation Res. 1, [4] KINGMAN, J. F. C. (1965) The heavy traffic approximation in the theory of queues. Proc. Symposium on Congestion Theory, W. L. Smith and W. E. Wilkinson (eds.), Univ. of North Carolina Monograph Series in Probability and Statistics. [5] NEWELL, G. F. (1965) Approximate methods for queues with application to the fixedcycle traffic light. SIAM Rev. 7, [6] Cox, D. R. AND MILLER, H. D. (1965) The Theory of Stochastic Processes. Chapter 5. John Wiley & Sons, New York. [7] Cox, D. R. AND SMrra, W. L. (1961). Queues. Methuen, London. [8] GAVER, D. P. (1968) Diffusion approximation and models for certain congestion problems. (In press).