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2 J. Appl. Prob. 5, (1968) Printed in Israel QUEUES WITH TIME-DEPENDENT ARRIVAL RATES II - THE MAXIMUM QUEUE AND THE RETURN TO EQUILIBRIUM G. F. NEWELL, Institute of Transportation and Traffic Engineering, University of California, Berkeley Abstract During a rushjhour, the arrival rate A(t) of customers to a service facility is assumed to increase to a maximum value exceeding the service rate j, and then decrease again. In Part I it was shown that, after A(t) has passed p, the expected queue E{X(t)} exceeds that given by the deterministic theory by a fixed amount, (0.95)L, which is proportional to the (-1/3) power of a(t) = da(t)/dt evaluated at time t = 0 when A(t) = p. The maximum of E(X(t)}, therefore, occurs when A(t) again is equal to p at time tl as predicted by deterministic queueing theory, but is larger than given by the deterministic theory by this same constant (0.95)L (provided tl is sufficiently large). It is shown here that the maximum queue, suptx(t) is, approximately normally distributed with a mean (0.95) (L + L1) larger than predicted by deterministic theory where L1 is proportional to the (-1/3) power of a(ti). We also investigate the distribution of X(t) at the end of the rush hour when the queue distribution returns to equilibrium. During the transition, the queue distribution is approximately a mixture of a truncated normal and the equilibrium distributions. These results are applied to a case where A(t) is quadratic in t. 1. Introduction In real life queueing situations it is quite common that customers arrive at a service facility at a time dependent rate 2(t). During a rush hour, A(t) increases to a maximum that exceeds the service capacity p and then subsides again. In Part I [1] we investigated the approximate distribution of queue length X(t) as a function of t, particularly over the transition region where 2(t) is increasing and passing through the saturation point 2(t) = p. We also determined the limit distributions for X(t) that will exist after the average queue has become so large as to have a negligible probability of being zero. It was assumed in I that A(t) increased approximately linearly with t during the transition period. The following is a continuation of Part I. We consider here some further properties of the same type of queueing situation but at some later period of time. Received 20 November

3 580 G. F. NEWELL In particular, we shall investigate (1) the distribution function of the maximum queue to occur during the rush hour, supx(t), which we expect will occur near the time when A(t) has passed through its maximum and is again approximately equal to yu; and (2) the evolution of the queue distribution after 2(t) has been less than yi long enough so that the queue has come back to zero again. As in I*, we will continue to identify t = 0 as the time when (t) = u for the first time. We also assume that dl(t)/dt remains nearly constant for It T, I(16a). Subsequently, however, d2(t)/dt starts to decrease. Eventually 2(t) reaches a maximum and then decreases. It passes through y again and continues to decrease until it becomes arbitrarily small. The maximum queue length will not occur until such time that the queue distribution is approximately normal with mean and variance given by 1(20), 1(30), and 1(32). The expected queue length (1) E{X(t)} ^,, A- a(u)du has its maximum at time t, when the time derivative of (1) vanishes, i.e. (2) - a(tl) = 2(tl) - y = 0. This is the same as would by given by the deterministic queueing theory, which differs only in that A = 0. In the deterministic queueing theory, one makes no distinction between the actual queue and the expected queue, so the maximum queue and the maximum expected queue are the same. In a stochastic model, however, the maximum queue is a random variable and the expectation of the maximum queue is, in general, larger than the maximum of the expectation of the queue. We would anticipate, though, that the maximum queue would occur at some time in the vicinity of the time tj. In the deterministic theory, the rush hour ends when the queue vanishes, i.e., at such time t2 when t2 (3) foa(u)du = 0. In the stochastic theory we must add a correction for the term A in (1), but, more important than this, we must also consider that the queue does not vanish at a deterministic time. It vanishes for the first time at some random time, and eventually the queue distribution approaches an equilibrium distribution (with a time dependent mean). * Equations from Part I will be designated by I(. ).

4 Queues with time-dependent arrival rates. II: The maximum queue The maximum queue In order to determine the distribution for the maximum queue, we first consider some properties of the function (4) G(x, t, Xo, to, z) = P{X(t)? x and X(rq) z for all r, for xo and x <z, to < t. to < q? t X(to) = x0o For z -+ oo, this function becomes the conditional distribution function for X(t) given that X(to) = xo; identified in 1(17) by the notation F(x, t xo, to). For x -+ z, G(z, t, Xo, to, z) considered as a function of z, is the distribution function for the maximum queue between times to and t, given that the queue is xo at time to, i.e., (5) G(z,t,xo, to, z) = P{ sup X(q)? zix(to) = xo}. to :5 t We assume here that for t sufficiently far away from t,, A(t) is small, and there is a negligible probability that the maximum queue over the whole real time axis - oo to + oo will occur outside some finite time interval around t1. Equivalently, we assume that (5) has a limit for t -+ + oo (6) G(z,oo, xo, to, z) = P{ sup X(q) < z X(to) = x o t?_,<oo which is also a proper distribution function (it has a limit 1 for z ) interpreted as the distribution function for the maximum queue after time to, given that it is xo at time to. Also if we take xo = 0 and let to -+ - oo we obtain a limit (7) G(z) = G(z, oo, O, - oo, z) = P{ sup X(q) < zix(- oo) = 0} - 00oo_!? oo which is a proper distribution function, interpreted as the distribution function of the maximum queue over the infinite time range. We are primarily concerned only with the evaluation of (7), but we must study some of the properties of (4), (5), and (6) as a means to this end. If we consider (4) as a function of the "final state" variables x and t, G is approximately a solution of the forward diffusion equation ag (ag b(t) 82G (8) - a -(-t) + the same equation as for F(x, t) in 1(2) except that G must satisfy the boundary conditions (9a) G(x, t, xo, to, z) -+ 0 for x - 0,

5 582 G. F. NEWELL ag (9b) (x, t, X, to, z) - 0 for x - z, 1 x>xo (9c) G(x, t, xo, to, z) 1~ < for ti to. 0 X < Xo Condition (9a) is the reflecting condition at x = 0, (9b) the absorbing condition at z, and (9c) the initial conditions. Considered as a function of the initial state variables xo and to, G is also approximately a solution of the backward diffusion equation [2], (10) G a G b(to) 02G (10) to a2(to) X2 the adjoint equation to (8), subject to the boundary conditions b(to) ag 2 ax0 (Ila) -a(to)g + 2 ( o -+0 for xo0-+0, (1lb) G(x,t,Xo, to, z) -0 for xo -+z, (I1c) G(x, t, Xo, to, z) -+for toit. Ox <xo Finally, one other important feature of (4) is that it has the semi-group property. For any to < t < tf (12) G(xf, tf, xo, to, z) = dxg(x, t, xo, to, z) G(xf, tf, x, t, z), i.e., the probability of X(r) going from x0 at time to to some state below xf at time tf,, is the probability of going from xo at time to to x at time t, and then from x to below xf at time tf, summed over all "intermediate states" x; all subject to X(ir) staying below z. We will not be able to solve the above equations exactly; the two boundary conditions at both x = 0 and x = z cause complications. To see what sort of approximations may be appropriate, however, consider a typical realization of X(r) such as illustrated in Figure 1. The mean queue represented by the broken line starts from some small value at negative times, rises to a maximum at time t1, and then comes back down. A typical realization may fluctuate about this mean, but on the scale of Figure 1, the fluctuations should be of relatively small amplitude. We are interested in evaluating G for values of z in the vicinity of this maximum. Now pick some time t, T < t < t1 but not too close to t1 and another time t, t, < tf < t, but not too close to either t1 or t2. The process X(q) is not likely to exceed z at any time r < t or r > tf, and is not likely to touch zero at any

6 Queues with time-dependent arrival rates. II: The maximum queue 583 Rea I ization Sx Menueue x to O t tf Time Figure 1 time ir with t < r < tf. Thus the time axis can be divided into regions where only one or the other boundary conditions at x = 0 or x = z are effective. Our procedure for estimating G(z) will be to approximate it by (13) G,(z) f dxg*(x, t)g*(x,t,z) analogous to (12) in which G*(x, t) - G(x, t, O, - oo, + oo) is the solution of (8) and (9) for initial conditions xo = 0 at to = - co and the absorbing barrier removed to + oo. Equivalently, it is the solution of (8) and (9) with condition (9b) replaced by G -+ 1 for x -+ + co. Thus (14) G*(x, t) = F(x, t), with F(x, t) the distribution function for X(t) as evaluated in I. The G*(x, t, z) is a solution of the backward equations (10) and (11) with initial coordinates x and t (instead of xo and to), and final coordinates xf and tf (instead of x and t). The final coordinates are in turn chosen to be xf = z and tf = + 00, but the condition (11a) is replaced by G*(x, t, z) -+ 1 for x -+ - oo, i.e., the reflecting condition at x = 0 is removed. If we were to imagine a hypothetical process Y(tr) which is the same as X(tr) for r < t but for r > t differs from X(r) in that the "queue" Y(r) may become negative without interruption of the service, then G,(z) is the distribution function for the maximum of Y(r) over tr> t. Obviously the maximum X(r) over - oo00 < < + oo00 is at least as large as the maximum of Y(r) over t < q < + 00 on two accounts; first because Y(tr)? X(tr) for all r and secondly because the maximum of X(tr) could occur for r < t. From this we conclude that Gt(z) is not only an approximation to G(z) over some range of t within (0, t1), it is also an upper bound G(z)? Gt(z) for all z and t.

7 584 G. F. NEWELL Also (15) G(z)? min G,(z). Finally, to evaluate G,(z) we must solve the backward diffusion equation for G*(x, t, z). Having removed the reflecting barrier at x = 0, we no longer have a preferred origin from which to measure the queue length. G*(x, t, z) is the conditional probability that the queue will never exceed z after time t, given that it was x at time t. Since the arrivals and departure processes, thus the changes in the queue length, do not depend upon the queue length, with the restriction against negative queues removed, G*(x, t, z) must be a function of t and z - x only, i.e., (16) G*(x, t, z) = G**(z - x, t). The G** is a solution of the equation (17)OG**(., t) = a(t)og**(, t) b(t) 02G(, t) at a 2 02 and the boundary conditions (18a) G**(O,t) = 0 (18b) G**(oo, t) = 1 (18c) G**(o, oo) = 1 for > 0. Condition (18a) implies that the maximum queue must be at least as large as the initial queue; (18b) implies that the maximum queue is a proper random variable; and (18c) implies that in the distant future the arrival rate will be so low that any initial queue is almost certain to decrease after time t. Equations (17) and (18) look very similar to Equations 1(2) and 1(6) of I. If we measure time backwards from tj, by making t' = t1 - t, (17) becomes (19) G**(, + t - t') = a(t a(t t') G**( -, t, )- t') + b(t, 2 - t') 2d2G(, t, - t') which differs from 1(2) only in that F(x, t) is replaced by G**(?, t - t'), the coefficients are evaluated at t1 - t' instead of t, and x, t are replaced by?, t'. Under the same correspondence, the boundary conditions (18) are also identical to 1(6). The time t1 was defined in (2) as a time when a(t) vanished, i.e., a(t - t') vanishes at t' = 0, just as in I, t = 0 was defined as the time when a(t) vanished for the first time. As in I, it is reasonable to assume that in some neighborhood of time t1 a, (t1 - t') varies linearly with t':

8 Queues with time-dependent arrival rates. II: The maximum queue 585 (20) a(tl - t') = p - 2(tl - t') ~ - aclt for some constant a, > 0 (the analogue of 1(14)). At time t1, A(t) is decreasing in t, therefore increasing in t'. Since the equations for G**(?, t) are identical in form to those for F(x, t) in I, we need only translate the properties of F(x, t) from I into corresponding properties for G**(?, t). As in I, there will be a characteristic time and length (21) T,= i P '04) and LL = (I+I )2/3, -. If (20) holds over a time t' of order TI, then G**(?, t) is approximately exponential in? for t' < 0, I1t' T,, i.e., for t-t1 >> T1, (22) G**(?, t) ~ 1 - exp [- 2a(t)?/b(t)]. There is a transition range I t' I = O(T,) where the maximum queue exceeds the value at time t by an amount of order L1. Finally for t' > T, the distribution G**(, t) is approximately normal with a mean (23) A1 - a(u)du and variance (24) B + ftb(u)du, with (25) A, ow(0.95)li and B - (0.3)L2E corresponding to 1(20), 1(30), and 1(32). Some of the qualitative properties of the above are intuitively reasonable. For large t, after the mean queue is decreasing, the peak queue after time t must occur either at time t or within a short time after t before the coefficients a(t) or b(t) have had time to change very much. The exponential distribution agrees with known solutions for constant a(t) and b(t). For t near tx we have again this peculiar dependence of queue length on the (-1/3) power of a, = da(t)/dt. It is clear that if a, is very large, the arrival rate will start to drop before the queue has had time to rise much above its given value at time t. Thus for a, -+ oo, the maximum queue will be arbitrarily close to its value at time t. But if a, is very small, the arrivals and departures stay nearly balanced for a long time. Eventually some fluctuation is likely to cause the queue to rise well above (or below) its value at time t.

9 586 G. F. NEWELL For t < t1, the queue will (on the average) still be increasing after time t and will be approximately normally distributed at time t1. The second terms of (23) and (24) are the mean and variance of the change in queue length from time t to t,. The corrections A1 and B, can be interpreted as a measure of the difference between the distribution of the peak queue and the distribution of the queue at time 1, when the mean queue is largest. If t, > T+ T1, and we choose t so that t > T and t -t > T1, then G**((, t), G*(x, t, z) from (16), and G*(x, t) from (14) are all normal distributions. Gt(z) is a convolution of two normal distributions which is also normal with a mean equal to the sum of the means of G*(x, t, z) and G*(x, t), (26) A + A - a(u)du = (0.95)(L + L1) + f[(u) - p ]du, and a variance equal to the sum of the variances, (27) B + B + b(u)du -(0.3)(L2 + L2) + [IhA(u) + It]duu. This Gt(z) is our estimate of the distribution function G(z) for the maximum queue. Note that for t in the range specified above, G,(z) does not depend upon t. In (26), the integral represents the peak queue as given by the deterministic queueing model. The term (0.95) (L + L,) is the excess due to stochastic effects. In (27) the integral represents the variance of the uninterrupted arrival and departure processes during the time 0 to t,. The first terms of (26) and (27) are valid estimates only if these terms are small compared with the second terms. Thus (26), in particular, is justified only if the deterministic theory already is a reasonable first approximation. The deterministic queueing models, however, do not define even approximately the conditions for their own validity. From (26) and some of the qualitative properties of the errors made in its derivation, one can also infer a converse. If the first term of (26) is not small compared with the second, then neither the deterministic model nor (26) is valid. 3. Return to equilibrium In the section of I dealing with undersaturated conditions, we saw that for any time to, there are a characteristic length Lo and characteristic time To (28) Lo = b(to)/a(to), To = b(to)la2(to) which represent respectively the scale of the equilibrium queue length and the relaxation time (the time it would take a queue of order Lo to become distributed approximately as the equilibrium distribution). In the vicinity of the second transition time t1, the characteristic length and time L, and T1, (21), are of the same order as Lo and To evaluated at time

10 Queues with time-dependent arrival rates. II: The maximum queue 587 to ~ t1 + T1. This time to represents about the end of the transition region and is characterized by this condition that the equilibrium scales of length and time are comparable with those for the transition. As t continues to increase past t, + T1, a(t) also increases, causing Lo and To both to decrease. For t - tj > T,, we would expect that Lo '< L, and To < <T,. In particular, we would expect this to apply by the time t ~ t2, (3), when the large queue built up during the time 0 to t, has been cut down to a small value again. Until there is a significant probability that the queue has vanished, the queue distribution will be normally distributed as in 1(20). The mean queue length is decreasing toward zero but the variance of the queue (under previously assumed properties) will be large compared with Li, which in turn is large compared with Lo2 for t - t2. Thus as t increases toward t2, the distribution of queue length is spread over a range which is very large compared with the local characteristic scale of length Lo. Suppose that for some time t near t2, we wish to evaluate 1 - F(x,t) = P(X(t) > x) for some x large compared with Lo. If there were no reflecting barrier at x = 0, this probability would be given by the normal distribution 1(20). But there is a negligible probability that any realization of X(ir) could pass through the state X(r) = 0 for tj < t and subsequently reach a value larger than x at time t, with or without reflection from the barrier. If there were no reflection, then according to (22), the maximum queue length that would be realized after the queue has vanished is of order Lo. If there is reflection, then a queue of length comparable with Lo at time ri would become distributed like the equilibrium distribution within a time of order To, and would have a very small probability of reaching a value large compared with Lo at any time t > ij. Thus for x > Lo, the distribution 1 - F(x, t) must be the tail of the normal distribution 1(20) in the absence of a barrier, and arise from queue realizations which have not yet hit the barrier. The probability mass that has not yet hit the barrier x = 0 by time t, is moving toward the barrier at an average "velocity" of Lo/To. The amount that will hit the barrier for the first time during a time interval of duration To must be of the order of the amount of probability mass in some queue range of width Lo. Since the probability mass that has not yet hit the barrier is approximately normally distributed on a scale of queue length large compared with Lo, only a small amount of this probability lies in any queue interval of width Lo and can hit the barrier during a time interval of duration To. Any non-negligible probability not contained in the normally distributed tail at time t, must therefore have hit the barrier for the first time at a time earlier than t by an amount appreciably larger than To, and will have become distributed approximately as the prevailing equilibrium distribution by time t.

11 588 G. F. NEWELL The queue distribution at time t must therefore be given approximately by (29) Fo(x, t) + F(O, t)exp[-2a(t)x/b(t)] for x > 0, 1 for x <0, where Fo(x, t) is the normal distribution function of 1(20) Fo(x, t) = (x -- m(t) (29a) m(t) A - f a(u)du a2(t)~ B + fb(u)du. The first part of (29) is the probability mass that has not yet reached x = 0; the second part is that which has hit the barrier and gone into the equilibrium distribution. The first part has a relatively long tail compared with the second. The queue distribution will approach the equilibrium distribution as Fo(O, t) approaches 1, i.e., at such time t3 that the mean of the normal distribution is negative by about one standard deviation u(t) (30) fo a(u)du {,fb(u)du}/ (since the right hand side of (30) is assumed to be large compared with L, we can neglect the A and B in (29a)). The mean queue length during this last transition will be Sm ( t) a 2(t) E{X(t)}.^ m(t) (u(t) ( + (27r) ( exp2-- /2 12U(t) 2 (31) ( m1 + m(t) b(t) for t -tx > T1. 4. Example (a(t) 2a(t) Here and in I we have considered the properties of a queue when the arrival rate 2(t) increased to a maximum and then decreased again. We assumed that 2(t) was approximately linear in t over the transition regions near t = 0 and t = t but otherwise )(t) could have a more or less arbitrary analytic form. To illustrate the application of the theory, we consider in more detail an example in which 2(t) is a quadratic function of t, the simplest analytic form consistent with the assumed shape for 2(t). We also assume that b(t)= b is constant over the time range of interest.

12 Queues with time-dependent arrival rates. II: The maximum queue 589 If as in I we choose t = 0 as the time when a(t) = 0 and we rescale the time and length coordinates with units T and L, we can assume that a(t) and b(t) have the form (32) a(t) = - t + t2/y, b(t) = 1 for some constant y. In order that a(t) be nearly linear over the first transition (over a range I t = 0(1)), we must have y > 1. The deterministic queueing theory predicts a queue rtp t2 t 3y ( 2 - /y)dt =,for 0 <t < (33) E{X(t)} f , for t < 0 or t > 3y/2. This is shown by the dotted curve of Figure 2. The maximum queue occurs at t = y and has a value of y2/6. Figure 2 is drawn for y = 5 which is already easily large enough to qualify as "large compared with 1". 7 l x, \ 6tY=5 4 I 1 6/ \ 6/ 0.95 Pt- % 6 / \I CM= \ 3 /\ l ". o~' - 'j\(t) 3 \ - ~ Equilibrium\ Mean7` -i i 3 4 Figure 2 Time \2(3, ) Also shown in Figure 2 is the mean queue as given by the diffusion approximation (solid line). Near t = 0, E(X(t)} is of order 1. It is copied directly from Figure 2 of I. From t ~ 1 until t ~ 3y/2, the mean queue exceeds (33) by A ~ Near t = 3y/2, the mean queue is evaluated from (29). The broken line curves of Figure 2 show the mean + the standard deviation of X(t). The variance of X(t) is determined directly from Figure 3 of I for t = 0(1). For t < 1 until t ~ 3y/2, it is

13 590 G. F. NEWELL r2(t)= b(u)du = t. Near t = 3y/2, it must be found from (19). At the maximum point t = y, the standard deviation of X(t) is ( y)1/2 ~ y1/2; as t approaches 3y/2, the standard deviation is about (3y/2)1/2 The queue distribution goes into its equilibrium distribution at time about t3 given by (30), i.e., when t t2 1/2 3y 3y 2 t or t3-12 3y 2t 2 t3/2 (3y/2)1/2 The final equilibrium distribution has a mean b(t) 1 2 2(at) 2a(t2) 3y Figure 2 shows that the first transition region near t = 0 ends when the mean - a(t) starts to drift away from zero. The last transition region near 3y/2 starts when the mean - oa(t) gets close to zero again signalling the start of reflections off the boundary. At this time the mean of the distribution (31) also starts to deviate from the mean of the normal distribution, m1(t). From a mathematical point of view, it is rather surprising that an equation of such simple form as the diffusion equation, involving only one parameter y, has a solution, important features of which involve such a variety of powers of the parameter. The units have been chosen so that queue lengths and the transition time near t = 0 are of order 1, i.e., y0. The maximum queue is of order y2 and it occurs at a time of order y'. The fluctuations in the queue as measured by the standard deviation are typically of order y'./ The transition time to the equilibrium state near time t2 is Of order y-' 1/2 and the final equilibrium distribution has a queue length of order y-. There are six different powers of y in this description. It is clear that the form of the solution is meaningful only if y > 1. If y becomes less than 1, the relative order of all these powers of y would be reversed and nothing would make much sense anymore. We shall see in Part III that for y < 1 one obtains some solutions of a different type involving still other powers of y, namely y1/5 and,2/5. References [1] NEWELL, G. F. (1968) Queues with time-dependent arrival rates. I - The transition through saturation. J. Appl. Prob. 5, [2] Cox, D. R. AND MILLER, H. D. (1965) The Theory of Stochastic Processes. Chapter 5. John Wiley & Sons, New York.

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