Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at


 Robyn Lyons
 1 years ago
 Views:
Transcription
1 Queues with TimeDependent Arrival Rates: II. The Maximum Queue and the Return to Equilibrium Author(s): G. F. Newell Source: Journal of Applied Probability, Vol. 5, No. 3 (Dec., 1968), pp Published by: Applied Probability Trust Stable URL: Accessed: :31 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at JSTOR is a notforprofit 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 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 TIMEDEPENDENT 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 timedependent 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 semigroup 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 timedependent 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 timedependent 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 tt1 >> 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 timedependent 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 nonnegligible 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 timedependent 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 t312 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 timedependent 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.
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at
Queues with TimeDependent Arrival Rates I:The Transition through Saturation Author(s): G. F. Newell Source: Journal of Applied Probability, Vol. 5, No. 2 (Aug., 1968), pp. 436451 Published by: Applied
More informationYour use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at
On the Estimation of the Intensity Function of a Stationary Point Process Author(s): D. R. Cox Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 27, No. 2 (1965), pp. 332337
More informationThe Econometric Society is collaborating with JSTOR to digitize, preserve and extend access to Econometrica.
On the Optimal Character of the (s, S) Policy in Inventory Theory Author(s): A. Dvoretzky, J. Kiefer, J. Wolfowitz Reviewed work(s): Source: Econometrica, Vol. 21, No. 4 (Oct., 1953), pp. 586596 Published
More informationBiometrika Trust. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika.
Biometrika Trust Discrete Sequential Boundaries for Clinical Trials Author(s): K. K. Gordon Lan and David L. DeMets Reviewed work(s): Source: Biometrika, Vol. 70, No. 3 (Dec., 1983), pp. 659663 Published
More informationYour use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at
Biometrika Trust Robust Regression via Discriminant Analysis Author(s): A. C. Atkinson and D. R. Cox Source: Biometrika, Vol. 64, No. 1 (Apr., 1977), pp. 1519 Published by: Oxford University Press on
More informationYour use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at
Biometrika Trust Some Simple Approximate Tests for Poisson Variates Author(s): D. R. Cox Source: Biometrika, Vol. 40, No. 3/4 (Dec., 1953), pp. 354360 Published by: Oxford University Press on behalf of
More informationEach copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
On Runs of Residues Author(s): D. H. Lehmer and Emma Lehmer Source: Proceedings of the American Mathematical Society, Vol. 13, No. 1 (Feb., 1962), pp. 102106 Published by: American Mathematical Society
More informationYour use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at
Regression Analysis when there is Prior Information about Supplementary Variables Author(s): D. R. Cox Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 22, No. 1 (1960),
More informationMathematical Association of America
Mathematical Association of America http://www.jstor.org/stable/2975232. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at. http://www.jstor.org/page/info/about/policies/terms.jsp
More informationThe Periodogram and its Optical Analogy.
The Periodogram and Its Optical Analogy Author(s): Arthur Schuster Reviewed work(s): Source: Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character,
More informationWeek 9 Generators, duality, change of measure
Week 9 Generators, duality, change of measure Jonathan Goodman November 18, 013 1 Generators This section describes a common abstract way to describe many of the differential equations related to Markov
More informationEach copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
On the Bound for a Pair of Consecutive Quartic Residues of a Prime Author(s): R. G. Bierstedt and W. H. Mills Source: Proceedings of the American Mathematical Society, Vol. 14, No. 4 (Aug., 1963), pp.
More informationBiometrika Trust. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika.
Biometrika Trust An Improved Bonferroni Procedure for Multiple Tests of Significance Author(s): R. J. Simes Source: Biometrika, Vol. 73, No. 3 (Dec., 1986), pp. 751754 Published by: Biometrika Trust Stable
More information16. Working with the Langevin and FokkerPlanck equations
16. Working with the Langevin and FokkerPlanck equations In the preceding Lecture, we have shown that given a Langevin equation (LE), it is possible to write down an equivalent FokkerPlanck equation
More informationEach copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
Measuring the Speed and Altitude of an Aircraft Using Similar Triangles Author(s): Hassan Sedaghat Source: SIAM Review, Vol. 33, No. 4 (Dec., 1991), pp. 650654 Published by: Society for Industrial and
More informationYour use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at
American Society for Quality A Note on the Graphical Analysis of Multidimensional Contingency Tables Author(s): D. R. Cox and Elizabeth Lauh Source: Technometrics, Vol. 9, No. 3 (Aug., 1967), pp. 481488
More informationSince D has an exponential distribution, E[D] = 0.09 years. Since {A(t) : t 0} is a Poisson process with rate λ = 10, 000, A(0.
IEOR 46: Introduction to Operations Research: Stochastic Models Chapters 56 in Ross, Thursday, April, 4:55:35pm SOLUTIONS to Second Midterm Exam, Spring 9, Open Book: but only the Ross textbook, the
More informationYour use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at
Biometrika Trust Some Remarks on Overdispersion Author(s): D. R. Cox Source: Biometrika, Vol. 70, No. 1 (Apr., 1983), pp. 269274 Published by: Oxford University Press on behalf of Biometrika Trust Stable
More informationEach copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
Selecting the Better Bernoulli Treatment Using a Matched Samples Design Author(s): Ajit C. Tamhane Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 42, No. 1 (1980), pp.
More informationYour use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at
A Note on the Efficiency of LeastSquares Estimates Author(s): D. R. Cox and D. V. Hinkley Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 30, No. 2 (1968), pp. 284289
More informationEach copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
The Variance of the Product of Random Variables Author(s): Leo A. Goodman Source: Journal of the American Statistical Association, Vol. 57, No. 297 (Mar., 1962), pp. 5460 Published by: American Statistical
More informationYour use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at
A Quincuncial Projection of the Sphere Author(s): C. S. Peirce Source: American Journal of Mathematics, Vol. 2, No. 4 (Dec., 1879), pp. 394396 Published by: The Johns Hopkins University Press Stable URL:
More informationEach copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
A Generic Property of the Bounded Syzygy Solutions Author(s): Florin N. Diacu Source: Proceedings of the American Mathematical Society, Vol. 116, No. 3 (Nov., 1992), pp. 809812 Published by: American
More informationb) The system of ODE s d x = v(x) in U. (2) dt
How to solve linear and quasilinear first order partial differential equations This text contains a sketch about how to solve linear and quasilinear first order PDEs and should prepare you for the general
More informationSTOCHASTIC PROCESSES Basic notions
J. Virtamo 38.3143 Queueing Theory / Stochastic processes 1 STOCHASTIC PROCESSES Basic notions Often the systems we consider evolve in time and we are interested in their dynamic behaviour, usually involving
More informationBiometrika Trust. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika.
Biometrika Trust A Stagewise Rejective Multiple Test Procedure Based on a Modified Bonferroni Test Author(s): G. Hommel Source: Biometrika, Vol. 75, No. 2 (Jun., 1988), pp. 383386 Published by: Biometrika
More informationTHE HEAVYTRAFFIC BOTTLENECK PHENOMENON IN OPEN QUEUEING NETWORKS. S. Suresh and W. Whitt AT&T Bell Laboratories Murray Hill, New Jersey 07974
THE HEAVYTRAFFIC BOTTLENECK PHENOMENON IN OPEN QUEUEING NETWORKS by S. Suresh and W. Whitt AT&T Bell Laboratories Murray Hill, New Jersey 07974 ABSTRACT This note describes a simulation experiment involving
More informationInternational Biometric Society is collaborating with JSTOR to digitize, preserve and extend access to Biometrics.
400: A Method for Combining NonIndependent, OneSided Tests of Significance Author(s): Morton B. Brown Reviewed work(s): Source: Biometrics, Vol. 31, No. 4 (Dec., 1975), pp. 987992 Published by: International
More informationEach copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
Uncountably Many Inequivalent Analytic Actions of a Compact Group on Rn Author(s): R. S. Palais and R. W. Richardson, Jr. Source: Proceedings of the American Mathematical Society, Vol. 14, No. 3 (Jun.,
More information1.225 Transportation Flow Systems Quiz (December 17, 2001; Duration: 3 hours)
1.225 Transportation Flow Systems Quiz (December 17, 2001; Duration: 3 hours) Student Name: Alias: Instructions: 1. This exam is openbook 2. No cooperation is permitted 3. Please write down your name
More informationEach copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
Merging of Opinions with Increasing Information Author(s): David Blackwell and Lester Dubins Source: The Annals of Mathematical Statistics, Vol. 33, No. 3 (Sep., 1962), pp. 882886 Published by: Institute
More informationMind Association. Oxford University Press and Mind Association are collaborating with JSTOR to digitize, preserve and extend access to Mind.
Mind Association Response to Colyvan Author(s): Joseph Melia Source: Mind, New Series, Vol. 111, No. 441 (Jan., 2002), pp. 7579 Published by: Oxford University Press on behalf of the Mind Association
More informationThe Econometric Society is collaborating with JSTOR to digitize, preserve and extend access to Econometrica.
A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision Author(s): Kenneth O. May Source: Econometrica, Vol. 20, No. 4 (Oct., 1952), pp. 680684 Published by: The Econometric
More informationGENERALIZED ANNUITIES AND ASSURANCES, AND INTERRELATIONSHIPS. BY LEIGH ROBERTS, M.Sc., ABSTRACT
GENERALIZED ANNUITIES AND ASSURANCES, AND THEIR INTERRELATIONSHIPS BY LEIGH ROBERTS, M.Sc., A.I.A ABSTRACT By the definition of generalized assurances and annuities, the relation is shown to be the simplest
More informationYour use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at
A Simple NonDesarguesian Plane Geometry Author(s): Forest Ray Moulton Source: Transactions of the American Mathematical Society, Vol. 3, No. 2 (Apr., 1902), pp. 192195 Published by: American Mathematical
More informationEach copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
On the Probability of Covering the Circle by Rom Arcs Author(s): F. W. Huffer L. A. Shepp Source: Journal of Applied Probability, Vol. 24, No. 2 (Jun., 1987), pp. 422429 Published by: Applied Probability
More informationTHE INTERCHANGEABILITY OF./M/1 QUEUES IN SERIES. 1. Introduction
THE INTERCHANGEABILITY OF./M/1 QUEUES IN SERIES J. Appl. Prob. 16, 690695 (1979) Printed in Israel? Applied Probability Trust 1979 RICHARD R. WEBER,* University of Cambridge Abstract A series of queues
More informationXt i Xs i N(0, σ 2 (t s)) and they are independent. This implies that the density function of X t X s is a product of normal density functions:
174 BROWNIAN MOTION 8.4. Brownian motion in R d and the heat equation. The heat equation is a partial differential equation. We are going to convert it into a probabilistic equation by reversing time.
More informationMath 331 Homework Assignment Chapter 7 Page 1 of 9
Math Homework Assignment Chapter 7 Page of 9 Instructions: Please make sure to demonstrate every step in your calculations. Return your answers including this homework sheet back to the instructor as a
More informationDerivation of Itô SDE and Relationship to ODE and CTMC Models
Derivation of Itô SDE and Relationship to ODE and CTMC Models Biomathematics II April 23, 2015 Linda J. S. Allen Texas Tech University TTU 1 EulerMaruyama Method for Numerical Solution of an Itô SDE dx(t)
More informationStochastic Modelling Unit 1: Markov chain models
Stochastic Modelling Unit 1: Markov chain models Russell Gerrard and Douglas Wright Cass Business School, City University, London June 2004 Contents of Unit 1 1 Stochastic Processes 2 Markov Chains 3 Poisson
More informationEach copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
6625 Author(s): Nicholas Strauss, Jeffrey Shallit, Don Zagier Source: The American Mathematical Monthly, Vol. 99, No. 1 (Jan., 1992), pp. 6669 Published by: Mathematical Association of America Stable
More informationMathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly.
A Proof of Weierstrass's Theorem Author(s): Dunham Jackson Reviewed work(s): Source: The American Mathematical Monthly, Vol. 41, No. 5 (May, 1934), pp. 309312 Published by: Mathematical Association of
More informationON LARGE SAMPLE PROPERTIES OF CERTAIN NONPARAMETRIC PROCEDURES
ON LARGE SAMPLE PROPERTIES OF CERTAIN NONPARAMETRIC PROCEDURES 1. Summary and introduction HERMAN RUBIN PURDUE UNIVERSITY Efficiencies of one sided and two sided procedures are considered from the standpoint
More informationCHAPTER V. Brownian motion. V.1 Langevin dynamics
CHAPTER V Brownian motion In this chapter, we study the very general paradigm provided by Brownian motion. Originally, this motion is that a heavy particle, called Brownian particle, immersed in a fluid
More informationHITTING TIME IN AN ERLANG LOSS SYSTEM
Probability in the Engineering and Informational Sciences, 16, 2002, 167 184+ Printed in the U+S+A+ HITTING TIME IN AN ERLANG LOSS SYSTEM SHELDON M. ROSS Department of Industrial Engineering and Operations
More informationOperations Research, Vol. 30, No. 2. (Mar.  Apr., 1982), pp
Ronald W. Wolff Operations Research, Vol. 30, No. 2. (Mar.  Apr., 1982), pp. 223231. Stable URL: http://links.jstor.org/sici?sici=0030364x%28198203%2f04%2930%3a2%3c223%3apasta%3e2.0.co%3b2o Operations
More information6 Solving Queueing Models
6 Solving Queueing Models 6.1 Introduction In this note we look at the solution of systems of queues, starting with simple isolated queues. The benefits of using predefined, easily classified queues will
More informationQUEUING MODELS AND MARKOV PROCESSES
QUEUING MODELS AND MARKOV ROCESSES Queues form when customer demand for a service cannot be met immediately. They occur because of fluctuations in demand levels so that models of queuing are intrinsically
More informationTHEODORE VORONOV DIFFERENTIABLE MANIFOLDS. Fall Last updated: November 26, (Under construction.)
4 Vector fields Last updated: November 26, 2009. (Under construction.) 4.1 Tangent vectors as derivations After we have introduced topological notions, we can come back to analysis on manifolds. Let M
More informationPolymerization and force generation
Polymerization and force generation by Eric Cytrynbaum April 8, 2008 Equilibrium polymer in a box An equilibrium polymer is a polymer has no source of extraneous energy available to it. This does not mean
More informationQ = (c) Assuming that Ricoh has been working continuously for 7 days, what is the probability that it will remain working at least 8 more days?
IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2005, Professor Whitt, Second Midterm Exam Chapters 56 in Ross, Thursday, March 31, 11:00am1:00pm Open Book: but only the Ross
More informationComputational statistics
Computational statistics Markov Chain Monte Carlo methods Thierry Denœux March 2017 Thierry Denœux Computational statistics March 2017 1 / 71 Contents of this chapter When a target density f can be evaluated
More informationINFORMS is collaborating with JSTOR to digitize, preserve and extend access to Management Science.
On the Translocation of Masses Author(s): L. Kantorovitch Source: Management Science, Vol. 5, No. 1 (Oct., 1958), pp. 14 Published by: INFORMS Stable URL: http://www.jstor.org/stable/2626967. Accessed:
More information1 Markov decision processes
2.997 DecisionMaking in LargeScale Systems February 4 MI, Spring 2004 Handout #1 Lecture Note 1 1 Markov decision processes In this class we will study discretetime stochastic systems. We can describe
More informationContinuousTime Markov Chain
ContinuousTime Markov Chain Consider the process {X(t),t 0} with state space {0, 1, 2,...}. The process {X(t),t 0} is a continuoustime Markov chain if for all s, t 0 and nonnegative integers i, j, x(u),
More information8 Ecosystem stability
8 Ecosystem stability References: May [47], Strogatz [48]. In these lectures we consider models of populations, with an emphasis on the conditions for stability and instability. 8.1 Dynamics of a single
More informationThe American Mathematical Monthly, Vol. 104, No. 8. (Oct., 1997), pp
Newman's Short Proof of the Prime Number Theorem D. Zagier The American Mathematical Monthly, Vol. 14, No. 8. (Oct., 1997), pp. 7578. Stable URL: http://links.jstor.org/sici?sici=2989%2819971%2914%3a8%3c75%3anspotp%3e2..co%3b2c
More informationLecture 7: Simulation of Markov Processes. Pasi Lassila Department of Communications and Networking
Lecture 7: Simulation of Markov Processes Pasi Lassila Department of Communications and Networking Contents Markov processes theory recap Elementary queuing models for data networks Simulation of Markov
More informationYour use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at
The Interpretation of Interaction in Contingency Tables Author(s): E. H. Simpson Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 13, No. 2 (1951), pp. 238241 Published
More informationRecap. Probability, stochastic processes, Markov chains. ELECC7210 Modeling and analysis of communication networks
Recap Probability, stochastic processes, Markov chains ELECC7210 Modeling and analysis of communication networks 1 Recap: Probability theory important distributions Discrete distributions Geometric distribution
More informationThis is a Gaussian probability centered around m = 0 (the most probable and mean position is the origin) and the mean square displacement m 2 = n,or
Physics 7b: Statistical Mechanics Brownian Motion Brownian motion is the motion of a particle due to the buffeting by the molecules in a gas or liquid. The particle must be small enough that the effects
More informationNEW FRONTIERS IN APPLIED PROBABILITY
J. Appl. Prob. Spec. Vol. 48A, 209 213 (2011) Applied Probability Trust 2011 NEW FRONTIERS IN APPLIED PROBABILITY A Festschrift for SØREN ASMUSSEN Edited by P. GLYNN, T. MIKOSCH and T. ROLSKI Part 4. Simulation
More informationEach copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
Modalities in Ackermann's "Rigorous Implication" Author(s): Alan Ross Anderson and Nuel D. Belnap, Jr. Source: The Journal of Symbolic Logic, Vol. 24, No. 2 (Jun., 1959), pp. 107111 Published by: Association
More informationIntroduction to Queueing Theory with Applications to Air Transportation Systems
Introduction to Queueing Theory with Applications to Air Transportation Systems John Shortle George Mason University February 28, 2018 Outline Why stochastic models matter M/M/1 queue Little s law Priority
More informationLIMITS FOR QUEUES AS THE WAITING ROOM GROWS. Bell Communications Research AT&T Bell Laboratories Red Bank, NJ Murray Hill, NJ 07974
LIMITS FOR QUEUES AS THE WAITING ROOM GROWS by Daniel P. Heyman Ward Whitt Bell Communications Research AT&T Bell Laboratories Red Bank, NJ 07701 Murray Hill, NJ 07974 May 11, 1988 ABSTRACT We study the
More informationEcological Society of America is collaborating with JSTOR to digitize, preserve and extend access to Ecology.
Measures of the Amount of Ecologic Association Between Species Author(s): Lee R. Dice Reviewed work(s): Source: Ecology, Vol. 26, No. 3 (Jul., 1945), pp. 297302 Published by: Ecological Society of America
More informationReflected Brownian Motion
Chapter 6 Reflected Brownian Motion Often we encounter Diffusions in regions with boundary. If the process can reach the boundary from the interior in finite time with positive probability we need to decide
More informationYour use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at
Biometrika Trust The Use of a Concomitant Variable in Selecting an Experimental Design Author(s): D. R. Cox Source: Biometrika, Vol. 44, No. 1/2 (Jun., 1957), pp. 150158 Published by: Oxford University
More informationSTAT 380 Continuous Time Markov Chains
STAT 380 Continuous Time Markov Chains Richard Lockhart Simon Fraser University Spring 2018 Richard Lockhart (Simon Fraser University)STAT 380 Continuous Time Markov Chains Spring 2018 1 / 35 Continuous
More informationThe Review of Economic Studies, Ltd.
The Review of Economic Studies, Ltd. Oxford University Press http://www.jstor.org/stable/2297086. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at.
More information221A Lecture Notes Convergence of Perturbation Theory
A Lecture Notes Convergence of Perturbation Theory Asymptotic Series An asymptotic series in a parameter ɛ of a function is given in a power series f(ɛ) = f n ɛ n () n=0 where the series actually does
More informationHEAVYTRAFFIC EXTREMEVALUE LIMITS FOR QUEUES
HEAVYTRAFFIC EXTREMEVALUE LIMITS FOR QUEUES by Peter W. Glynn Department of Operations Research Stanford University Stanford, CA 943054022 and Ward Whitt AT&T Bell Laboratories Murray Hill, NJ 079740636
More informationE[X n ]= dn dt n M X(t). ). What is the mgf? Solution. Found this the other day in the Kernel matching exercise: 1 M X (t) =
Chapter 7 Generating functions Definition 7.. Let X be a random variable. The moment generating function is given by M X (t) =E[e tx ], provided that the expectation exists for t in some neighborhood of
More informationStochastic process. X, a series of random variables indexed by t
Stochastic process X, a series of random variables indexed by t X={X(t), t 0} is a continuous time stochastic process X={X(t), t=0,1, } is a discrete time stochastic process X(t) is the state at time t,
More informationLatent voter model on random regular graphs
Latent voter model on random regular graphs Shirshendu Chatterjee Cornell University (visiting Duke U.) Work in progress with Rick Durrett April 25, 2011 Outline Definition of voter model and duality with
More informationLECTURE #6 BIRTHDEATH PROCESS
LECTURE #6 BIRTHDEATH PROCESS 204528 Queueing Theory and Applications in Networks Assoc. Prof., Ph.D. (รศ.ดร. อน นต ผลเพ ม) Computer Engineering Department, Kasetsart University Outline 2 BirthDeath
More informationACM 116: Lectures 3 4
1 ACM 116: Lectures 3 4 Joint distributions The multivariate normal distribution Conditional distributions Independent random variables Conditional distributions and Monte Carlo: Rejection sampling Variance
More informationDISTRIBUTIONS FUNCTIONS OF PROBABILITY SOME THEOREMS ON CHARACTERISTIC. (1.3) +(t) = eitx df(x),
SOME THEOREMS ON CHARACTERISTIC FUNCTIONS OF PROBABILITY DISTRIBUTIONS 1. Introduction E. J. G. PITMAN UNIVERSITY OF TASMANIA Let X be a real valued random variable with probability measure P and distribution
More informationThe Performance Impact of Delay Announcements
The Performance Impact of Delay Announcements Taking Account of Customer Response IEOR 4615, Service Engineering, Professor Whitt Supplement to Lecture 21, April 21, 2015 Review: The Purpose of Delay Announcements
More informationREVIEW: Waves on a String
Lecture 14: Solution to the Wave Equation (Chapter 6) and Random Walks (Chapter 7) 1 Description of Wave Motion REVIEW: Waves on a String We are all familiar with the motion of a transverse wave pulse
More informationFigure 10.1: Recording when the event E occurs
10 Poisson Processes Let T R be an interval. A family of random variables {X(t) ; t T} is called a continuous time stochastic process. We often consider T = [0, 1] and T = [0, ). As X(t) is a random variable
More informationLecture for Week 2 (Secs. 1.3 and ) Functions and Limits
Lecture for Week 2 (Secs. 1.3 and 2.2 2.3) Functions and Limits 1 First let s review what a function is. (See Sec. 1 of Review and Preview.) The best way to think of a function is as an imaginary machine,
More informationA Simple Solution for the M/D/c Waiting Time Distribution
A Simple Solution for the M/D/c Waiting Time Distribution G.J.Franx, Universiteit van Amsterdam November 6, 998 Abstract A surprisingly simple and explicit expression for the waiting time distribution
More informationPaul Mullowney and Alex James. Department of Mathematics and Statistics University of Canterbury Private Bag 4800 Christchurch, New Zealand
REsONANCES IN COMPOUND PROCESSES Paul Mullowney and Alex James Department of Mathematics and Statistics University of Canterbury Private Bag 4800 Christchurch, New Zealand Report Number: UCDMS2006/14 DECEMBER
More informationCONTINUOUS STATE BRANCHING PROCESSES
CONTINUOUS STATE BRANCHING PROCESSES BY JOHN LAMPERTI 1 Communicated by Henry McKean, January 20, 1967 1. Introduction. A class of Markov processes having properties resembling those of ordinary branching
More information01 Harmonic Oscillations
Utah State University DigitalCommons@USU Foundations of Wave Phenomena Library Digital Monographs 82014 01 Harmonic Oscillations Charles G. Torre Department of Physics, Utah State University, Charles.Torre@usu.edu
More information2. As we shall see, we choose to write in terms of σ x because ( X ) 2 = σ 2 x.
Section 5.1 Simple OneDimensional Problems: The Free Particle Page 9 The Free Particle Gaussian Wave Packets The Gaussian wave packet initial state is one of the few states for which both the { x } and
More informationDetection of Influential Observation in Linear Regression. R. Dennis Cook. Technometrics, Vol. 19, No. 1. (Feb., 1977), pp
Detection of Influential Observation in Linear Regression R. Dennis Cook Technometrics, Vol. 19, No. 1. (Feb., 1977), pp. 1518. Stable URL: http://links.jstor.org/sici?sici=00401706%28197702%2919%3a1%3c15%3adoioil%3e2.0.co%3b28
More informationExamination paper for TMA4265 Stochastic Processes
Department of Mathematical Sciences Examination paper for TMA4265 Stochastic Processes Academic contact during examination: Andrea Riebler Phone: 456 89 592 Examination date: December 14th, 2015 Examination
More informationMatrices A(t) depending on a Parameter t. Jerry L. Kazdan
Matrices A(t depending on a Parameter t Jerry L. Kazdan If a square matrix A(t depends smoothly on a parameter t are its eigenvalues and eigenvectors also smooth functions of t? The answer is yes most
More information1. Find the solution of the following uncontrolled linear system. 2 α 1 1
Appendix B Revision Problems 1. Find the solution of the following uncontrolled linear system 0 1 1 ẋ = x, x(0) =. 2 3 1 Class test, August 1998 2. Given the linear system described by 2 α 1 1 ẋ = x +
More informationQueueing Theory II. Summary. ! M/M/1 Output process. ! Networks of Queue! Method of Stages. ! General Distributions
Queueing Theory II Summary! M/M/1 Output process! Networks of Queue! Method of Stages " Erlang Distribution " Hyperexponential Distribution! General Distributions " Embedded Markov Chains M/M/1 Output
More informationYour use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at
A Look at Some Data on the Old Faithful Geyser Author(s): A. Azzalini and A. W. Bowman Source: Journal of the Royal Statistical Society. Series C (Applied Statistics), Vol. 39, No. 3 (1990), pp. 357365
More informationUniversal examples. Chapter The Bernoulli process
Chapter 1 Universal examples 1.1 The Bernoulli process First description: Bernoulli random variables Y i for i = 1, 2, 3,... independent with P [Y i = 1] = p and P [Y i = ] = 1 p. Second description: Binomial
More informationDefinition 5.1. A vector field v on a manifold M is map M T M such that for all x M, v(x) T x M.
5 Vector fields Last updated: March 12, 2012. 5.1 Definition and general properties We first need to define what a vector field is. Definition 5.1. A vector field v on a manifold M is map M T M such that
More informationData analysis and stochastic modeling
Data analysis and stochastic modeling Lecture 7 An introduction to queueing theory Guillaume Gravier guillaume.gravier@irisa.fr with a lot of help from Paul Jensen s course http://www.me.utexas.edu/ jensen/ormm/instruction/powerpoint/or_models_09/14_queuing.ppt
More informationFigure 1: Doing work on a block by pushing it across the floor.
Work Let s imagine I have a block which I m pushing across the floor, shown in Figure 1. If I m moving the block at constant velocity, then I know that I have to apply a force to compensate the effects
More information1 The Observability Canonical Form
NONLINEAR OBSERVERS AND SEPARATION PRINCIPLE 1 The Observability Canonical Form In this Chapter we discuss the design of observers for nonlinear systems modelled by equations of the form ẋ = f(x, u) (1)
More information