Performance Evaluation Of Finite Horizon. Characteristics of Waiting Times In. (Max,Plus) Linear Queueing Networks

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1 Performance Evaluation Of Finite Horizon Characteristics of Waiting Times In (Max,Plus) Linear Queueing Networks Bernd Heidergott Vrije Universiteit Amsterdam and Tinbergen Institute, De Boelelaan 1105, 1081 HV Amsterdam, the Netherlands, Abstract We propose a new approach to performance evaluation of transient performance characteristics of stochastic (max,+) linear systems. Our motivating example is the transient waiting time in the G/G/1 queue. Numerical results will illustrate the performance of our method. Keywords: performance analysis, perturbation analysis, (max,+) algebra 1

2 1 Introduction A wide range of queueing systems enjoy the property that their sample path dynamic becomes linear when modeled in (max,+) algebra. Examples of such (max,+) linear queueing systems are the G/G/1-queue, networks of queues in tandem, Kanban systems, fork/join queues or any parallel and/or series composition made by these elements, see [12]. In this paper we introduce a novel approach to performance evaluation of finite horizon performance indicators of stochastic (max,+)-linear queueing systems, called variability expansion, to be explained in the following. Let {A(k)} be an i.i.d. sequence of square matrices over the (max,+) algebra and consider the (max,+) recurrence relation x(k +1) = A(k) x(k), k 0, (1) with x(0) = x 0, see the Appendix for basic definitions of (max,+) algebra. Our goal is to evaluate E[g(x(m))] for fixed m and given performance indicator g. To this end, we introduce a parameter and replace with probability 1 the random matrix A(k) in the above recursion by its mean. The parameter allows controlling the level of randomness in the system: letting go from 0 to 1 increases the level of stochasticity in the system. For example, =0 represents a completely deterministic system, whereas =1 represents the (fully) stochastic system (that is, the original one). Denote by {x (k)} the version of {x(k)}, for [0, 1].For =1, it holds that E[g(x (m))] = E[g(x(m))]. In order to evaluate E[g(x (m))] at =1, we develop E[g(x (m))] into a Taylor series with respect to at =0. For the sake of simplicity, we illustrate our approach 2

3 with the waiting time in the G/G/1 queue. Consider a G/G/1 queue with i.i.d. interarrival times {σ 0 (k) :k 1} and i.i.d. service times {σ 1 (k) : k 1}. Denote by σ 0 the mean interarrival time and by σ 1 the mean service time, and assume that ρ := σ 1 /σ 0 < 1. The system is initially empty and the waiting time of the k th customer, denoted by W (k), follows: W (k +1)=σ 1 (k) ( σ 0 (k + 1)) W (k) 0 = max( σ 1 (k) σ 0 (k +1)+W (k), 0), k 0, (2) with W (0) = 0 and σ 1 (0) = 0. Note that, for k =0, (2) gives W (1) = max( σ 1 (0) σ 0 (1), 0) = 0, which recovers that fact that the first customer arriving at an empty queue doesn t experience any waiting time. In order to write (2) as a homogeneous equation, such as (1), we set: A(k) = Let w(0) = (0, 0) and set σ 1(k) σ 0 (k +1) 0, k 0. ɛ 0 w(k +1) = A(k) w(k), k 0, then w(k) =(W (k), 0). In words, the first component of w(k) is the actual waiting time of the k th customer. Set g W (w(k)) = g(w (k)), then ( k 1 ) g(w (k)) = g W (w(k)) = g W A(j) w(0), k 1. 3 j=0

4 The deterministic variant of the system is obtained by replacing the random entries of A(k) by their means, that is, by considering the transition matrix A = σ 1 σ 0 0. ɛ 0 In order to construct a version of {W (k)} that combines deterministic transitions according to A with random ones according to {A(k)}, we proceed as follows. Let A(k) have distribution µ and denote the Dirac measure in x by δ x. Let D (k) R 2 2 max, for k N, be an i.i.d. sequence with distribution µ = µ +(1 )δ A. (3) In words, with probability, D (k) behaves like A(k), whereas, with probability (1 ), D (k) is equal to A. For [0, 1], we set w (k +1) = D (k) w (k), k 0, with w (0) = (0, 0). We call the transition from w (k) to w (k +1)deterministic if D (k) =A and stochastic otherwise. We write E [g W (w(m))] to indicate that the version is considered. For fixed m>0, the performance characteristic of the transient waiting time of the version is thus given by E [g W (w(m))], where E[g W (w(m))] = E 1 [g W (w(m))] and E 0 [g W (w(m))] = g W (A m w(0)). Let h N. Suppose that, for n h+1, the n th derivative of E [g W (w(m))] exists, where at a boundary point we define derivatives as a one sided limit. Specifically, we set lim 0 d n d n E [g W (w(m))] = 4 dn d n E 0[g W (w(m))],

5 then E[g(W (m))] = E 1 [g W (w(m))] the true expected performance characteristic of the m th waiting time is given by where E[g(W (m))] = R h+1 (m) = 1 h! 1 h n=0 0 1 d n n! d E 0[g n W (w(m))] + R h+1 (m) (4) (1 t) h dh+1 d h+1 E [g W (w(m))] dt. =t d As we will show in this paper, n E d n 0 [g W (w(m))] can be evaluated by making n of the m transitions stochastic and letting the other transitions be deterministic. Hence, the variability of dn E d n 0 [g W (w(m))] increases with the order of the derivative, which motivates the name variability expansion for the Taylor series in (4). Variability expansion has been first introduced in [14] but no proofs were given. An application of variability expansion to approximate computation of multi dimensional integrals stemming from a problem in control theory can be found in [8]. Taylor series expansions of (max,+) linear queueing systems are an area of active research. The approach dominant in the literature is that of light traffic approximations for stationary waiting times in open (max,+) linear queuing systems with Poisson arrival stream. More specifically, let W i denote the i th component of the vector of stationary waiting times in an open queueing system with Poisson λ arrival stream. The pioneering paper on light traffic expansions for E[W i ] is [5], where sufficient conditions for the existence of the light traffic approximation for E[W i ] with repsect to λ are established and the (first) elements of the Taylor series are computed analytically. These results have been extended in [7] to E[f(W i )], where f belongs to the class of performance measures F, where h F 5

6 if h :[0, ) [0, ) and h(x) cx ν for x 0 and ν N. In [1] expansions are obtained for E[f(W i,w j )], for f :[0, ) 2 [0, ) with f(x, y) cx ν 1 x ν 2 for x, y 0 and ν 1,ν 2 N. In [2, 3], explicit expressions are given for the moments, Laplace transform and tail probability of the waiting time of the n th customer. Furthermore, starting with these exact expressions for transient waiting times, exact expressions for moments, Laplace transform and tail probability of stationary waiting times in a certain class of (max,+) linear systems with deterministic service are computed. Variability expansion differs from the above approaches: it is neither a light traffic approximation nor does it assume that the arrival stream if of Poisson λ type. The paper is organized as follows. Section 2 shows that the n th order derivative of E [g W (w(m))] exists and derives an explicit formula for the derivative. The main result is presented in Section 3, where the Taylor series in explicitly calculated. A recursive scheme for computing coefficient of the Taylor series is given in Section 4. Numerical examples are presented in Section 5. 2 Higher order derivatives of transient waiting times The distribution of D (k), defined in (3), is differentiable with respect to : µ = µ δ A. (5) In words, the derivative of µ with respect to can be written as difference between the probability measures µ and δ A. Such a representation is called weak 6

7 derivative in the literature; see [16]. Notice that higher order derivatives of µ with respect to are zero. The theory developed in [16] has been applied to (max,+) linear systems in [13]. To motivate the following theorem, denote the m fold independent product of µ by µ m = µ µ µ }{{} m times For m =2, straightforward calculation yields ) = ) )) (µ µ ((µ +(1 )δ A (µ +(1 )δ A ) ) ) ) = (µ δ A (µ +(1 )δ A + (µ +(1 )δ A (µ δ A ) ) = (µ δ A µ + µ (µ δ A,. in shorthand notation: ( µ µ ) = µ µ + µ µ. It is easily seen that (µ m ) = m 1 l=0 µ m l 1 µ µl, (6) where µ k has to be disregarded whenever k =0. Inserting µ δ A for µ in (6) yields ) = m 1 (µ µ l=0 ( µ m l 1 µ µ l µm l 1 δ A µ l ). (7) Notice that the difference between (6) and (7) is that (6) involves a singed measure, namely µ, whereas the elements of the sum in (7) are differences between 7

8 products of probability measures. Hence, the expression in (7) can be interpreted as a stochastic experiment. For a second order derivative, we compute as follows d 2 ( ) ) µ d 2 µ )= ((µ µ ) = (µ µ + µ µ =µ µ + µ µ, where we last equality stems from the fact the fact that higher order derivatives of µ are zero. Notice that d 2 µ /d 2 =0. For general m we obtain, d 2 m 2 d 2 (µm )=2 m 1 l 1 =0 l 2 =l 1 +1 µ m l 2 1 µ µ l 2 l 1 1 µ µ l 1, (8) for m 2 and zero otherwise. Inserting (5) and rearranging terms yields d 2 ( ) m 2 µ d 2 µ =2 m 1 l 1 =0 l 2 =l 1 +1 m 2 +2 m 1 l 1 =0 l 2 =l 1 +1 m 2 2 l 1 =0 l 2 =l 1 +1 m 2 2 m 1 m 1 l 1 =0 l 2 =l 1 +1 µ m l 2 1 µ m l 2 1 µ m l 2 1 µ m l 2 1 µ µ l 2 l 1 1 µ µ l 1 δ A µ l 2 l 1 1 δ A µ l 1 µ µ l 2 l 1 1 δ A µ l 1 δ A µ l 2 l 1 1 µ µ l 1. In words, the positive parts are obtained from substituting both occurrences of µ in (8) either by µ (the positive part of µ ) or by δ A (the negative part of µ ). The negative parts are obtained from substituting µ at one occurrence by µ and at the other occurrence by δ A. The second order derivative already shows the complex combinatorial struc- 8

9 ture and the alternating sign of terms typical for higher order derivatives of products of measures. The extension of the above results to higher order derivatives is straightforward and, following the above line of argument, we arrive at the following result: d n d n µm = n! m n m n+1 l 1 =0 l 2 =l 1 +1 m 1 l n=l n 1 +1 µ m ln 1 µ µ µl 2 l 1 1 µ µl 1, for n m and zero otherwise. For 0 l 1 < l 2 < < l n m 1 let I[l 1,...,l n ; m] denote the set of i =(i 0,...,i m 1 ) { 1, 0+1} m such that i k =0, for k {l 1,l 2,...,l n }, and n i lk =1. k=1 For i I[l 1,...,l n ; m], we define i through i k = i k for k l n and i ln = i l n, where we write i k for the kth element of i. In words, i is obtained from i through changing the sign of the last non-zero element in i. Inserting (5) for µ and separating the positive and negative part then yields d n d n µm m n = n! m n+1 l 1 =0 l 2 =l 1 +1 i I[l 1,...,l n;m] m 1 l n=l n 1 +1 ( m k=1 µ (i k) m k=1 µ (i k ) ). (9) Let D denote the set of measurable mappings g : R J max R such that E [g W (w(m))] and g(a x), for any x R J, are finite for any [0, 1]. For 9

10 i { 1, 0, 1}, we set D (i) (k) = D (k) for i =0 A(k) for i =1 Let µ (0) = µ, µ (1) = µ and µ ( 1) A for i = 1. = δ A. Hence, D (i) (k) is distributed according to µ (i), for i { 1, 0, 1}. With this notation we write, for example, µ = µ(1) µ ( 1). Theorem 1 Let n m, the n th order derivative of E [ g W (w(m)) ] is given by d n d E [g n W (w(m))] m n m n+1 = n! l 1 =0 l 2 =l 1 +1 i I[l 1,...,l n;m] m 1 l n=l n 1 +1 i I[l 1,...,l n;m] [ ( m 1 E g W k=0 [ ( m 1 E g W D (i k) (k) w(0) k=0 )] D (i k ) (k) w(0) )], whereas the n th derivative is zero for n>m. Proof: Note that E [g W (w(m))] = g w (a 0,a 1,...,a m 1 )[µ m ](da 0,da 1,...,da m 1 ) and E [g W (w(m))] depends on only through µ m. Since µm is a polynomial in we may put the differentiation operator under the integration sing for any g D. 10

11 Inserting (9) for the n th derivative of µ yields d n g d n w (a 0,a 1,...,a m 1 )[µ m ](da 0,da 1,...,da m 1 ) [ ] d n = g w (a 0,a 1,...,a m 1 ) (da 0,da 1,...,da m 1 ) = n! m n m n+1 l 1 =0 l 2 =l 1 +1 i I[l 1,...,l n;m] i I[l 1,...,l n;m] m 1 l n=l n 1 +1 d n µm g w (a 0,a 1,...,a m 1 ) g w (a 0,a 1,...,a m 1 ) [ m 1 k=0 [ m 1 k=0 µ (i k) µ (i k ) ] (da 0,da 1,...,da m 1 ) ] (da 0,da 1,...,da m 1 ). By definition, D (i k) (k) has distribution µ (i k). Switching from the above measure representation to random variable view concludes the proof of the theorem. Letting tend to zero, those D (i k) (k) in the expression for the nth derivative in Theorem 1 for which i k =0converge in total variation to A. In the following, explicit representations of the first three derivatives of E [g W (w(m))] at =0are given. For 0 j<m, set V g (m; j) =E [ g W ( A m j 1 A(j) A j w(0) )] and V g (m) =g W ( A m w(0) ). Then d d E 0[g W (w(m))] = m 1 j=0 ( ) V g (m; j) V g (m). (10) 11

12 In the same vein, set, for 0 j 1 <j 2 <m, V g (m; j 1,j 2 )=E [ g W ( A m j 2 1 A(j 2 ) A j 2 j 1 1 A(j 1 ) A j 1 w(0) )], then d 2 d 2 E 0[g W (w(m))] =2 m 2 m 1 j 1 =0 j 2 =j 1 +1 ( ) V g (m; j 1,j 2 )+V g (m) V g (m; j 1 ) V g (m; j 2 ). For the third element set, for 0 j 1 <j 2 <j 3 <m, ( V g (m; j 1,j 2,j 3 )= E [g W A m j3 1 A(j 3 ) A j 3 j 2 1 )] A(j 2 ) A j 2 j 1 1 A(j 1 ) A j 1 w(0) and the third order derivative is obtained from d 3 d E 0[g 3 W (w(m))] m 3 m 2 =6 m 1 j 1 =0 j 2 =j 1 +1 j 3 =j 2 +1 ( V g (m; j 1,j 2,j 3 )+V g (m; j 1 )+V g (m; j 2 )+V g (m; j 3 ) ) V g (m; j 1,j 2 ) V g (m; j 2,j 3 ) V g (m; j 1,j 3 ) V g (m). The derivatives can be verbally described as follows. The factor is n! when the n th order derivative is evaluated. The outer summation ranges over all possible combinations of marking n out of m transitions. The inner sum ranges over all possible combinations of letting the n marked transitions be either stochastic or not. The sign of an element in the inner sum is given by 1 to the power of the number of deterministic substitutions among the n marked transitions. Proceeding as above, we can define factors V g (m; j 1,...,j k ), for 1 k m. The n th order derivative of E 0 [g W (w(m))] is then given through 12

13 V g (m; j 1,...,j k ), for 1 k n. Let V g (m, k) = m k m k+1 j 1 =0 j 2 =j 1 +1 m 1 j k =j k 1 +1 V g (m; j 1,...,j k ), for k m, and V g (m, 0) = V g (m). The term V g (m, l) yields the total effect of making l out of m transitions stochastic. For the n th derivative of E[g W (w(m))] we mark in total n transitions and out of which l are stochastic. Hence, there are m l n l possibilities of reaching at (m l) deterministic transitions provided that there l stochastic ones, and we obtain d n d E 0[g n W (w(m))] n = n! m l ( 1) n l V g (m, l). l=0 n l Assume that E 0 [g W (w(m))] is at least htimes differentiable. Inserting the above expression into the Taylor series and rearranging terms gives h n E[ g(w (m)) ] m l ( 1) n l V g (m, l) n=0 l=0 n l h h = m l ( 1) n l V g (m, l) l=0 n=l n l with = h C(h, m, l) V g (m, l), (11) l=0 C(h, m, l) = 1 (h l)! h (j m), (12) j=l+1 where we set the product to one for l = h; see [8] for details. 13

14 3 Variability Expansion Theorem 1 not only shows that all higher order derivatives of E [g W (w(m))] exists for any Θ, more importantly it shows that the (m +1) st derivative of E [g W (w(m))] equals zero. Hence, the remainder term in (4) equals zero for h>mand E [g W (w(m))] is thus analytical on R. Corollary 1 For any [0, 1], E [g W (w(m))] can be developed into a Taylor series in whose domain of convergence is [0, 1]. In particular, the remainder term for Taylor series in (4) equals zero for h>m. By Theorem 1, the resulting Taylor series approximation of degree h =3for is given by E[ g(w (m)) ] d d E 0[g W (w(m))] d d E 0[g 2 W (w(m))] d E 0[g 3 W (w(m))], where the approximation becomes exact for m 3. Inserting the expression for the n th order derivative into (11) and rearranging terms gives h E[ g(w (m)) ] = with C(h, m, l) as in (12). n=0 h = l=0 d 3 n m l ( 1) n l V g (m, l) n l h m l ( 1) n l V g (m, l) n l l=0 n=l h C(h, m, l) V g (m, l), h m, l=0 14

15 We conclude this section we some considerations of the value of Taylor series expansion as a tool for numerical approximation. Fix x 0 R and > 0, let f :[x 0,x 0 + ] R be an (n +1)times continuously differentiable mapping on [x 0,x 0 + ](where differentiability at the boundary has to interpreted as one sided differentiability). Then it holds that n m d m f(x 0 + ) = m! dx m f(x) +R n+1 (x 0, ) (13) m=0 x=x0 R n+1 (x 0, ) = 1 x0 + (x 0 + t) n dn+1 n! x 0 dx n+1 f(x) dt. (14) x=t The radius of convergence of a Taylor series, denoted by r(x 0 ), is the largest such that the R n+1 (x 0, ) on the righthand side of (13) tends to zero as ntends to. Let the Taylor series expansion for f at x 0 have radius of convergence r(x 0 ) > 0. The expression in (13) equals f for at least those x 0 +, with r(x 0 ), for which the remainder term R n+1 (x 0, ) tends to 0 as n tends to. Does this mean that increasing the degree of the Taylor polynomial improves the accuracy of the approximation? If d m f/x m 0for all m except at most finitely many, the answer is yes. This stems from the fact that, for n N, n f(x m d m 0 + ) m! dx m f(x) m=0 x=x0 m d m = m! dx m f(x) m=n+1 x=x0 m d m m! dx m f(x) m=n+1 x=x0 =: H n+1. Existence of the Taylor series implies that H n 0. Observe that R n (x 0, ) H n. Hence, the error in predicting f(x 0 + ) by a Taylor polynomial of degree n is 15

16 at most H n, where H n is a monotone decreasing sequence. Thus, with growing n, R n+1 (x 0, ) essentially decreases. However, this line of argument does not apply if d m f/dx m =0for m N, for some finite N. In this case, it is not ruled out that the accuracy of the approximation decreases with increasing the degree of the Taylor polynomial provided the degree is smaller than N. For example, it can happen that R n (x 0, ) <R n+1 (x 0, ) for n N 1 and R n+1 (x 0, ) = 0 for n N. In such a case, increasing the degree of the Taylor polynomial may even decrease the quality of the approximation. In this case we speak of deceiving analyticity. 4 Computation of the Taylor Series Expansion The coefficients of the Taylor series enjoy a recursive structure which can be exploited when calculating the series. In the following we will discuss this in more detail where the key observation is that a stochastic transition only contributes to the overall derivative if the waiting time introduced by that stochastic transition doesn t die out before the following stochastic transition occurs. For 0 i 1 < i 2 < i h < m, let W (m; i 1,i 2,...,i h ) denote the m th waiting time in the system with deterministic transitions except for transitions i 1,i 2,,i h. Let W( ) be the projection onto the first component of the vector w(k) and introduce the variables W [m; i] =W(A m i 1 A(i) A i w(0)), 0 i m, 16

17 W [m; i 1,i 2 ]=W(A m i 2 1 A(i 2 ) A i 2 i 1 1 A(i 1 ) A i 1 w(0)), for 0 i 1 <i 2 <mand W [m; i 1,i 2,i 3 ]=W(A m i 3 1 A(i 3 ) A i 3 i 2 1 A(i 2 ) A i 2 i 1 1 A(i 1 ) A i 1 w(0)), for 0 i 1 <i 2 <i 3 <m. In addition to that, set W [m] =W(A m w(0)). Consider the G/G/1 queue with mean interarrival time σ 0 and mean service time σ 1, and set c = σ 1 σ 0. Then, c is the so called drift of the random walk {W (k)} and because we have assumed that the system is stable, i.e. σ 1 /σ 0 < 1, the drift is negative. This can be phrased by saying that a deterministic transition decreases the amount of work present at the server by c. Denote the density of the interarrival times by f A and the density of the service times by f S and assume that f A and f S have support (0, ). We now turn to the computation of the derivative of E[g(W (m))] with respect to, see (10). First, notice that V g (m) =g(w(a m w(0))). For m>0, it is easily checked that V g (m; i)=g(0) P (W [m; i] =0) +E[1 W [m;i]>0 g(w [m; i])] = 0 a (m i 1)c +g(0) P (W [m; i] =0), g(s a +(m i 1)c) f S (ds) f A (da) 17

18 where P (W [m; i] =0) = a (m i 1)c 0 0 f S (ds) f A (da) and 1 W [m;i]>0 denotes the indicator mapping for the event {W [m; i] > 0}, that is, 1 W [m;i]>0 =1if W [m; i] > 0 and otherwise zero. We now turn to the second order derivative. For 0 i 1 <i 2 <m, W [i 2 ; i 1 ] > 0 describes the event that a stochastic transition at i 1 generated a workload at the server that hasn t been completely worked away until transition i 2. With the help of this event we can compute as follows V g (m; i 1,i 2 )=E [ g(w [m; i 1,i 2 ]) ] = E [ 1 W [i2 ;i 1 ]>01 W [m;i1,i 2 ]>0 g(w [m; i 1,i 2 ]) ] +E [ 1 W [i2 ;i 1 ]=01 W [m;i1,i 2 ]>0 g(w [m; i 1,i 2 ]) ] +E [ 1 W [i2 ;i 1 ]>01 W [m;i1,i 2 ]=0 g(w [m; i 1,i 2 ]) ] +E [ 1 W [i2 ;i 1 ]=01 W [m;i1,i 2 ]=0 g(w [m; i 1,i 2 ]) ]. On the event {W [i 2 ; i 1 ]=0} the effect of the first stochastic transition dies out before transition i 2. By independence, E [ 1 W [i2 ;i 1 ]=01 W [m;i1,i 2 ]>0 g(w [m; i 1,i 2 ]) ] = E [ 1 W [i2 ;i 1 ]=01 W [m;i2 ]>0 g(w [m; i 2 ]) ] = P ( W [i 2 ; i 1 ]=0 ) E [ 1 W [m;i2 ]>0 g(w [m; i 2 ]) ] and E [ 1 W [i2 ;i 1 ]=01 W [m;i1,i 2 ]=0 g(w [m; i 1,i 2 ]) ] = g(0)p ( W [i 2 ; i 1 ]=0 ) P ( W [m; i 2 ]=0 ). 18

19 Moreover, it is easily checked that E [ 1 W [i2 ;i 1 ]>01 W [m;i1,i 2 ]=0 g(w [m; i 1,i 2 ]) ] = g(0) P ( W [i 2 ; i 1 ] > 0 W [m; i 2 ]=0 ). We obtain V g (m; i 1,i 1 ) as follows: V g (m; i 1,i 2 )=E [ 1 W [m;i1,i 2 ]>0 1 W [i2 ;i 1 ]>0 g(w [m; i 1,i 2 ]) ] + E [ 1 W [m;i2 ]>0 g(w [m; i 2 ]) ] P ( W [i 2 ; i 1 ]=0 ) + g(0) (P ( W [i 2 ; i 1 ]=0 ) P ( W [m; i 2 ]=0 ) + P ( W [i 2 ; i 1 ] > 0 W [m; i 1,i 2 ]=0 )), where the expressions in the product on the right hand side in the above formula have already been calculated in the process of computing the first order derivative. In other words, in order to compute the second order derivative only m(m +1)/2 terms have to be computed, namely E[1 W [m;i1,i 2 ]>0 1 W [i2 ;i 1 ]>0 g(w [m; i 1,i 2 ]) ] for 0 i 1 <i 2 <m. These terms can be computed as follows: E [ 1 W [i2 ;i 1 ]>0 1 W [m;i1,i 2 ]>0 g(w [m; i 1,i 2 ]) ] = 0 0 a 1 +a 2 (m i 1 2)c 0 g(s 1 + s 2 a 1 a 2 +(m i 1 2)c) f S (s 2 )f S (s 1 )f A (a 2 )f A (a 1 ) ds 2 ds 1 da 2 da 1 a1 +a 2 (m i 1 2)c a 1 (i 2 i 1 1)c g(s 1 + s 2 a 1 a 2 +(m i 1 2)c) a 1 +a 2 s 1 (m i 1 2)c f S (s 2 )f S (s 1 )f A (a 2 )f A (a 1 ) ds 2 ds 1 da 2 da 1. Setting g =1and adjusting the boundaries of the integrals, we can compute from the above equations the probability of the event {W [i 2 ; i 1 ] > 0 W [i 3 ; i 1,i 2 ]=0}. 19

20 For the third order derivative the computations become more cumbersome. To abbreviate the notation, we set h i (s 1,s 2,s 3,a 1,a 2,a 3 ) = g(s 1 + s 2 + s 3 a 1 a 2 a 3 +(m i 3)c) f S (s 3 )f S (s 2 )f S (s 1 )f A (a 3 )f A (a 2 )f A (a 1 ) 20

21 and we obtain E [ 1 W [i2 ;i 1 ]>01 W [i3 ;i 1,i 2 ]>01 W [m;i1,i 2,i 3 ]>0 g(w [m; i 1,i 2,i 3 ]) ] = a 1 +a 2 +a 3 (m i 1 3)c a1 +a 2 (i 3 i 1 2)c h i1 (s 1,s 2,s 3,a 1,a 2,a 3 ) ds 3 ds 2 ds 1 da 3 da 2 da 1 a 1 (i 2 i 1 1)c + 0 a 1 +a 2 +a 3 (m i 1 3)c s a1 +a 2 (i 3 i 1 2)c h i1 (s 1,s 2,s 3,a 1,a 2,a 3 ) ds 3 ds 2 ds 1 da 3 da 2 da 1 a 1 (i 2 i 1 1)c a1 +a 2 +a 3 (m i 1 3)c s 1 a 1 +a 2 (i 3 i 1 2)c s 1 a 1 +a 2 +a 3 s 1 s 2 (m i 1 3)c h i1 (s 1,s 2,s 3,a 1,a 2,a 3 ) ds 3 ds 2 ds 1 da 3 da 2 da 1 21

22 a1 +a 2 +a 3 (m i 1 3)c a 1 +a 2 (i 3 i 1 2)c a 1 +a 2 +a 3 (m i 1 3)c s 1 0 h i1 (s 1,s 2,s 3,a 1,a 2,a 3 ) ds 3 ds 2 ds 1 da 3 da 2 da a1 +a 2 +a 3 (m i 1 3)c a 1 +a 2 (i 3 i 1 2)c a1 +a 2 +a 3 (m i 1 3)c s 1 0 a 1 +a 2 +a 3 (m i 1 3)c s 1 s 2 h i1 (s 1,s 2,s 3,a 1,a 2,a 3 ) ds 3 ds 2 ds 1 da 3 da 2 da 1. Following the line of argument used for the second derivative, the third order derivative can now be expressed as a combination of the above variables together with the variables already computed while calculating the first and second order derivatives. The precise formula is: [ V g (m; i 1,i 2,i 3 )=E ] 1 W [i2 ;i 1 ]>01 W [i3 ;i 1,i 2 ]>01 W [m;i1,i 2,i 3 ]>0 g(w [m; i 1,i 2,i 3 ]) [ ] P ( W [i 2 ; i 1 ]=0 ) + E 1 W [i3 ;i 2 ]>01 W [m;i2,i 3 ]>0 g(w [m; i 2,i 3 ]) [ ] + E 1 W [m;i3 ]>0 g(w [m; i 3 ]) (P ( W [i 2 ; i 1 ] > 0 W [i 3 ; i 1,i 2 ]=0 ) + g(0) P ( W [m; i 1,i 2,i 3 ]=0 ), +P ( W [i 2 ; i 1 ]=0 ) P ( W [i 3 ; i 2 ]=0 )) 22

23 where P ( W [m; i 1,i 2,i 3 ]=0 ) = P ( W [i 2 ; i 1 ]=0 ) P ( W [i 3 ; i 2 ] > 0 W [m; i 3,i 2 ]=0 ) +P ( W [i 2 ; i 1 ] > 0 W [i 3 ; i 1,i 2 ] > 0 W [m; i 3,i 2 ]=0 ) +P ( W [i 2 ; i 1 ]=0 ) P ( W [i 3 ; i 2 ]=0 ) P ( W [m; i 3 ]=0 ) +P ( W [i 2 ; i 1 ] > 0 W [i 3 ; i 1,i 2 ]=0 ) P ( W [m; i 3 ]=0 ). 5 Numerical Examples Consider g = id, that is, g(w (m)) = W (m), m 0. Note that ρ<1implies ( that V g (m) =g W A m w(0) ) = 0. Direct computation of E[W (m)] involves performing an m fold integration over a complex polytope. In contrast to this, the proposed variability expansion allows to build an approximation of E[W (m)] out of terms that involve h fold integration with h<m(below we have taken h =2, 3). This reduces the complexity of evaluating E[W (m)] considerably. To illustrate the performance of the variability expansion, we applied our approximation scheme to the transient waiting time in a stable (that is, ρ<1) M/M/1 queue and D/M/1 queue, respectively. 5.1 The M/M/1 Queue Table 1 lists values for the Taylor polynomial of degree h =2for various traffic loads. The values in brackets are the true values. For h =2, we are performing two stochastic transitions and a naive approximation of E[W (m)] is given through V id (2; 0, 1) = E[W (2)]. To illustrate the influence of h, we also evaluated the 23

24 Taylor polynomial of degree h = 3. See Table 2 for numerical results. For the true values we refer to Table 1. Here, the naive approximation is given by V id (3; 0, 1, 2) = E[W (3)]. Insert Table1 here. Insert Table2 here. Comparing the numerical values listed in Table 1 and Table 2, it turns out that for ρ<0.5 the Taylor polynomial of degree 3 provides a good approximation for the transient waiting time. However, the quality of the approximation decreases with increasing time horizon at ρ =0.3. Forρ 0.5, the approximation works well only for relatively small time horizons (m 10). It is worth noting that in heavy traffic (ρ =0.9) the quality of the approximation decreases when the third order derivative is taken into account. The erratic behaviour of the approximation at ρ =0.9 is best illustrated by the negative values for m =20and m =50. However, for m =5, the approximation still works well. To summarize, the quality of the approximation decreases with growing traffic intensity when the time horizon increases. Comparing Table 1 and Table 2, one notes that the outcome of the Taylor series approximation can be independent of the time horizon m. For example, at 24

25 ρ =0.1, the values of the Taylor polynomial do not vary in m. This stems from the fact that for such a small ρ the dependence of the m th waiting time on waiting times W (m k), k 5, is negligible. Hence, allowing transitions m k, k 5, to be stochastic doesn t contribute to the outcome of E[W (m)], which is reflected by the true values as well. The quality of the approximation decreases for growing ρ and growing h. This stems from the fact that convergence of the Taylor series is forced by the fact that the n th derivative of E [W (m)] jumps to zero at n = m. As described in Section 3 in the Appendix, in such a situation, the quality of the approximation provided by the Taylor polynomial may worsen through increasing h as long as h m. See Section 3 where the phenomenon of deceiving analyticity is discussed. The numerical values were computed with the help of a computer algebra program. The calculations were performed on a Laptop with Intel Pentium III processor and the computation times are listed in Table 3. Insert Table3 here. Note that the computational effort is independent of the traffic rate and only influenced by the time horizon. The table illustrates that the computational effort for computing the first two elements of the Taylor polynomial grows very slowly in m, whereas the computational effort for computing the first three elements of the Taylor series increases rapidly in m. This indicates that computing higher degree Taylor polynomials will suffer from high computational costs. 25

26 5.2 The D/M/1 Queue Table 4 lists values for the Taylor polynomial of degree h =2and h =3, respectively, for various traffic loads. Specifically, the upper values refer to the Taylor polynomial of degree h =2, the values in the second row are those for the Taylor series of degree h =3, and the values in brackets are the true values (which stem from intensive simulation). For the naive approximation, the values V id (2; 0, 1) are the upper values and the values V id (3; 0, 1, 2) are listed on the second row. Insert Table4 here. Tabel 4 shows the same behaviour of the variability expansions as already observed for the M/M/1 queue. Like for the M/M/1 queue, the quality of the approximation decreases with growing traffic intensity when the time horizon increases. The numerical values were computed with the help of a computer algebra program. The calculations were performed on a Laptop with Intel Pentium III processor and the computation times are listed in Table 5. Due to the fact that the interarrival times are deterministic, calculating the elements of the variability expansions for the D/M/1 queue requires less computation time than for the M/M/1 queue, see Table 3. Insert Table5 here. 26

27 Conclusion We presented a numerical scheme for approximately computing performance characteristics of transient waiting times in (max,+) linear queueing systems. We illustrated the numerical properties of our method with the M/M/1 queue and the D/M/1 queue. For traffic load less than 0.5 the methods produced reliable results. In high traffic situations however the error is prohibitive. Moreover, the phenomenon of deceiving analyticity was encountered and increasing the degree of the Taylor polynomial didn t necessarily lead to an improvement of the approximation. A thorough analysis of this phenomenon is topic of further research. We applied variability expansion to simple queueing systems and extension to networks of queues is topic of further research. In principle it is clear what has to be done. A straightforward extension of the proposed algorithm, however, will be numerically infeasible. Fortunately, the derivatives enjoy strong recursive properties and exploiting these inherent relations between derivatives may lead to a more efficient representation of the coefficients of the Taylor series. Appendix This section provides a brief introduction to (max,+) algebra. For an extensive discussion of the (max,+) algebra and similar structures we refer to [4] and [10]. An early reference is [9]. A historical overview on the beginnings of the (max,+) theory can be found in [11]. The (max,+) algebra is usually introduced as follows. Let ɛ = and let us denote by R ɛ the set R {ɛ}. For elements a, b R ɛ we define the operations 27

28 and by a b = max(a, b) and a b = a+b, where we adopt the convention that for all a R max(a, ) = max(,a)=aand a+( ) = +a =. The set R ɛ together with the operations and is called the (max,+) algebra and is denoted by R max. In particular, ɛ is the neutral element for the operation and absorbing for, that is, for all a R ɛ a ɛ = ɛ. The neutral element for is e =0. For matrices A R I J and B R J K the matrix product A B is defined in the usual way as follows: J (A B) ik = A ij B jk. (15) j=1 Example 1 Consider an open tandem network consisting of J FCFS single servers. Let queue 0 represent an external arrival stream of customers. Each customer who arrives at the system has to pass through queues 1 to J, and then leaves the system. For simplicity, we assume that the system starts empty. We let x j (k) denote the time of the kth service completion at station j and denote the kth service time at j by σ j (k) (where σ 0 (k) has to be interpreted as the kth interarrival time). In particular, we let x 0 (k) denote the kth arrival epoch at the system. The time evolution of the system can then be described by a (J +1) dimensional vector x(k) =(x 0 (k),...,x J (k)) following the recursion x(k +1)=A(k) x(k), k 0, with x(0) = (e,...,e), where the matrix A(k) is given by i (A(k)) ij = σ l (k +1) (16) l=j for J i j 0 and otherwise ɛ. Let W j (k) be the time the kth customer arriving at the network spends in the system until leaving node j, then W j (k) = x j (k) x 0 (k). 28

29 Let A be a random element in R I J ɛ defined on some probability space (Ω, A,P). We call A integrable if E[ A ij 1 Aij >ɛ ] < for all 1 i I and 1 j J. In other words, A is integrable if all entries A ij are absolutely integrable on the event that they are finite. In case P (A ij = ɛ) =1,we set E[A ij ]=. 29

30 References [1] Ayhan, H. and Baccelli, F. Expansions for joint Laplace transforms for stationary waiting times in (max,+) linear systems with Poisson input. QUESTA, 37: , [2] Ayhan, H. and Seo, D. Tail probability of transient and stationary waiting times in (max,+) linear systems. IEEE Trans. Aut. Cont., 47: , [3] Ayhan, H. and Seo, D. Laplace transform and moments of waiting times in Poisson driven (max,+) linear systems. QUESTA, 37: , [4] Baccelli, F., Cohen, G., Olsder, G. J., and Quadrat, J. P. Synchronization and Linearity. John Wiley and Sons, (this book is out of print and can be accessed via the max-plus web portal at [5] Baccelli, F. and Schmidt, V. Taylor series expansions for Poisson driven (max,+) linear systems. Annals Appl. Prob., 6: , [6] Baccelli, F., Hasenfuß, S., and Schmidt, V. Transient and stationary waiting times in (max,+) linear systems with Poisson input. QUESTA, 26: , [7] Baccelli, F., Hasenfuß, S., and Schmidt, V. Expansions for steady state characteristics of (max,+) linear systems. Com. in Statistcs: Series Stochastic Models, 14:1 24, [8] van den Boom, T., De Schutter, B., and Heidergott, B. Complexity reduction in MPC for stochastic max-plus-linear systems by variability expansion. In 30

31 Proceedings of the 41st IEEE Conference on Decision and Control, pages , Las Vegas, Nevada, December [9] Cuninghame Green, R.A. Minimax algebra. vol. 166 of Lecture Notes in Economics and Mathematical Systems. Springer Verlag, Berlin, [10] Cuninghame Green, R.A. Minimax algebra and applications. In: Advances in Imaging and Electron Physics 90 (P.W. Hawkes, Ed.). Academic Press, New York, [11] Gaubert, S. Methods and applications of (max,+) linear algebra. In Proceedings of the STACS 1997, Lecture Notes in Computer Science, vol Springer (this report can be accessed via the WEB at [12] Heidergott, B. A characterization for (max,+) linear queueing systems. QUESTA, 35: , [13] Heidergott, B. A differential calculus for random matrices with applications to (max,+) linear stochastic systems. Mathematics of Operations Research, 26: , [14] Heidergott, B. Variability expansion for performance characteristics of (max,plus)-linear systems. In Proceedings of the International Workshop on DES, Zaragoza, Spain, pages IEEE Computer Society, [15] Jean Marie, A. Waiting time distributions in Poisson driven deterministic systems. Technical report no. 3083, INRIA Sophia Antipolis,

32 [16] Pflug, G. Optimization of Stochastic Models. Kluwer Academic, Boston,

33 Table 1: Approximating the expected m th waiting time in a M/M/1 queue via a Taylor polynomial of degree 2 m ρ =0.1 ρ =0.3 ρ =0.5 ρ =0.7 ρ = [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] naive analytical (n = )

34 Table 2: Approximating the expected m th waiting time in a M/M/1 queue via a Taylor polynomial of degree 3 m ρ =0.1 ρ =0.3 ρ =0.5 ρ =0.7 ρ = naive Table 3: CPU time (in seconds) for computing Taylor polynomials of degree h for time horizon m in a M/M/1 queue m h=2 h=

35 Table 4: Approximating the expected m th waiting time in a D/M/1 queue via a Taylor polynomial of degree 2 and 3, respectively. m ρ =0.1 ρ =0.3 ρ =0.5 ρ =0.7 ρ = [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] naive

36 Table 5: CPU time (in seconds) for computing Taylor polynomials of degree h for time horizon m in a D/M/1 queue m h=2 h=