N.G.Bean, D.A.Green and P.G.Taylor. University of Adelaide. Adelaide. Abstract. process of an MMPP/M/1 queue is not a MAP unless the queue is a
|
|
- Dustin Daniels
- 5 years ago
- Views:
Transcription
1 WHEN IS A MAP POISSON N.G.Bean, D.A.Green and P.G.Taylor Department of Applied Mathematics University of Adelaide Adelaide 55 Abstract In a recent paper, Olivier and Walrand (994) claimed that the departure process of an MMPP/M/ queue is not a MAP unless the queue is a stationary M/M/ queue. They also conjectured that the result extends to MAP/PH/ queues. In the rst part of this paper we show that their proof has an algebraic error, which leaves the above question open. There is also a more fundamental problem with Olivier and Walrand's proof. In order to discuss the problem, it is essential to be able to determine from its generator when a stationary MAP is a Poisson process. This is not discussed in Olivier and Walrand (994), nor does it appear to have been discussed in the literature. This deciency is remedied in the second part of this paper, where we use ideas from non-linear ltering theory to give a characterisation as to when a stationary MAP is a Poisson process. INTRODUCTION. A Markovian Arrival Process, (MAP) is a process which counts transitions of a nite state Markov Chain. It is possible for a process which counts transitions of an innite state Markov Chain to be statistically equivalent to a MAP. For example, consider an M/M/ queue, with arrival rate > and service rate >, which is modelled by a Markov chain x = fx t ; t g on the state space Z +, where x t represents the number in the queue at time t. This Markov chain
2 has the following innite transition rate matrix : : : ( + ) : : : ( + ) : : : ; () where the number in the queue x, increases as the row number of the matrix. In the situation where <, the queue is positive recurrent and, under stationary conditions, the point process of occurrence of transitions (n+; n); n, is known (see Burke (956)) to be a Poisson process of rate, which is of course a trivial MAP. A natural question to ask is whether a similar property holds for other queues. That is, for more general queues, does there exist a nite state Markov chain and a set of transitions for which the counting process of the transitions is identical to the departure process of the original queue. Olivier and Walrand (994) presented an argument to show that there exists no such nite state chain for an MMPP/M/ queue and conjectured that this is also true for a MAP/M/ queue. Unfortunately there is an algebraic error in the argument of Olivier and Walrand, which we point out in Section 3, and so the question of whether the output of an MMPP/M/ queue can be a MAP still remains open. There is also a more fundamental problem when addressing this question, which was not discussed in Olivier and Walrand (994). Since it is possible for the arrival process of a MAP/M/ queue to be Poisson, but with a possibly complicated description, and since we know that the output of such a queue is a MAP, (as mentioned above, it is Poisson) it is essential to be able to tell from it's generator when a MAP is, in fact, Poisson. 2 MARKOV ARRIVAL PROCESSES (MAP s). We generalise the arrival process for the transition rate matrix in () by relaxing the requirement that the inter-arrival times be negative exponentially distributed. This is achieved by adding auxiliary states or phases to the arrival process and associating with each arrival, certain phase changes. The Markovian simplicity is still preserved since the sojourn times within phases of the auxiliary process are still negative exponentially distributed. We dene a two state Markov chain (x; y), where x represents the number in the queue at time t and y represents the phase of the arrival process at time t. Following the notation of Neuts (98) and letting the number of phases be m, we get the following block matrix form for the conservative rate matrix of the MAP/M/ queue. Q = B A A 2 A A A 2 A A : (2)
3 When the queue is empty, the matrix B governs the transitions of the arrival process which do not correspond to an arrival and A governs those transitions which do. When the, queue is occupied, the matrix A 2 = I mm governs departures, A governs arrivals and A = B A 2 governs those transitions which do not correspond to an arrival or a departure. Note that is a conservative rate matrix so that S = A + B = A + A + A 2 (3) S = : (4) As in Neuts (98) we assume that Q denes an irreducible, regular Markov chain. Necessary conditions for this are that the m m matrices B and A are nonsingular. Hence this Markov chain has at most, one stationary distribution such that Q =. This stationary distribution has a matrix-geometric form and is given by = [ ; R; R 2 ; : : : ; R x ; : : :] ; where R is the minimal non-negative solution to the matrix quadratic equation R 2 A 2 + RA + A = (5) and is the unique positive solution to the system of equations (B + RA 2 ) = and (I R) = : (6) 2. PH-RANDOM VARIABLES AND PH-RENEWAL PROCESSES. Any continuous distribution on [; ) which can be obtained as the distribution of time until absorption in a continuous-time nite-space Markov chain which has a single absorbing state into which absorption is certain, is said to be of Phase-type. Consider the following Markov chain with m+ states, initial probability vector (; m+ ) and transition rate matrix " # T T Q = ; where T is a non-singular m m matrix with T ii <, T ij for all i 6= j and T is an m vector such that T +T =. The conditional probabilities r j (t) that the process is in state j at time t with initial conditions (r (); r 2 (); : : : ; r m ()) = r() = (; m+ ) satisfy the dierential equations which have solution dr(t) = r(t) Q; dt r(t) = (; m+ ) e Qt :
4 The conditional probability vector v(t) that the process is still in one of the states ; : : : ; m at time t is given by Thus v(t) = e T t : F (t) = e T t ; is the probability distribution of time until absorption into state m +. This is classied as a Ph-type distribution with representation (; T ). If we consider the m + st state of the Ph-random variable as an instantaneous state, in that we instantaneously restart the process using the probability vector, then the process consisting of absorption epochs is a Ph-renewal process with representation (; T ). 3 THE QUESTION OF A MAP OUTPUT FROM A STATIONARY MAP/M/ QUEUE. We use the techniques of non-linear ltering, as given in Walrand (988). Consider a Markov chain x = fx t ; t g having a countably innite state space X, with transition rate matrix Q. For i 6= j 2 X, let Q (i; j) Q(i; j); for i 2 X let and let Q (i; i) < ; Q = Q Q : The transitions with rates Q are observed, while those in Q are hidden. Let J t count the number of observed transitions up to time t, and let t(k) = P fx t = kjj t g; so that t(k) is the probability of being in state k at time t conditioned by the number of observed jumps up to time t. Also let t be the row vector t = f t(k); k 2 Xg: A Markov chain with rate matrix Q is a MAP process if and only if there exists a nite state Markov chain with rate matrix Q and corresponding matrices Q and Q such that Q = Q + Q and t = t Q = t Q = t ; for all t 2 R + ; where is a column of 's of the appropriate dimension.
5 A necessary condition for the above to hold is given in Olivier and Walrand (994) to be t (Q ) y (Q ) x = t (Q ) y (Q ) x ; for all x; y : (7) Olivier and Walrand (994) considered the special case of an MMPP/M/ queue and from the necessary condition given in equation (7), derived the following for all x,! x! = I Q! x+ Q (I R) Q! x A ; (8) where is dened by equation (6) and is the stationary distribution of the conservative generator Q. Recalling the denition of the matrix S given in equation (3), we let = (I R), which is the stationary distribution admitted by S, so that S =. Using this and equation (4), Olivier and Walrand erroneously concluded that B! x =! x A + S = A! x : (9) >From this point they argued by contradiction that no nite state space equivalent Markov chain dened by Q could exist. Equation (9) however is incorrect. For example, a value of x = 3 will yield! 3! 3 A + S A = A 6= A! 3! + A SA 3 3. A COUNTER-EXAMPLE TO OLIVIER AND WALRAND'S ASSERTION. It is, in fact possible for equation (8) to be satised for a non-trivial MAP/M/ queue. Consider the case where the arrival process is a Ph-renewal process with the inter-event time distribution dened by an initial distribution and transition matrix B ( that is a Ph-renewal process with representation (; B )). Note that A can be written as A = B : () Let x be the stationary probability vector in the original queue that a departure leaves the system empty and the phase of the arrival process in state i, for i 2 f; : : : ; mg. Then if the (m + ) (m + ) matrices " # " # Q B A = and Q = ; () x ( x )
6 are substituted into equation (8), it is satised. The stationary solution for the conservative generator Q = (Q + Q ), in this case is given by = ( ; ) ; where is the m-vector dened previously in equation (6). It must be noted however that equation (8) is necessary but not sucient for the output process to be equivalent to a MAP. A necessary and sucient condition is given in the next section in equation (2). As a result, the question of whether the output process from an MMPP/M/ queue can be a MAP is still an open question, as is the similar question for that of any MAP/M/ queue or a MAP/PH/ queue. 4 WHEN IS A STATIONARY MAP POISON The question of when a stationary MAP is Poisson was not discussed in Olivier and Walrand (994). This also appears to be the case in the literature, even though it is a very natural question. We take up this problem using the techniques of nonlinear ltering. From Walrand (988) we see that two point processes dened by Q = Q + Q and Q = Q + Q have the same nite dimensional distributions if and only if for any given initial distributions of states and respectively e Q t Q e Q t 2 : : : Q e Q t k Q = e Q t Q e Q t 2 : : : Q e Q t k Q e Q t Q e Q t 2 : : : Q e Q t k e Q t Q e Q t 2 : : : Q ye Q t k for all k ; t k 2 [; ): (2) Until now we have only considered the case where Q is an innite matrix and Q is a nite matrix, however equation (2) can be used to compare any two point processes. Therefore for this section we rewrite equation (2) for the case where the right-hand side is a Poisson process of rate > and the left-hand side is a MAP. e Q t Q e Q t 2 : : : Q e Q t k Q e Q t Q e Q t 2 : : : Q e Q t k = for all k ; t k 2 [; ) ; (3) where Q = Q + Q. Any process Q = Q + Q which is equivalent to a Poisson process of rate >, must satisfy equation (3). We will consider the matrix Q in it's spectral form, concentrating on the the situation where Q has distinct eigenvalues and can be written Q = i= i r i l i : (4) Note that trivially, any MAP which has the same arrival rate in every phase of the arrival process will satisfy equation (3) which can be seen by noticing that for such a MAP, Q =, giving a Poisson process of rate =. This result is
7 not aected by the initial distribution and corresponds to the case where is a right eigenvector of the matrix Q and as Q is conservative, also a right eigenvector of Q and Q. 4. A PHASE (PH) RANDOM VARIABLE. We consider initially the equivalence of a Ph-random variable to a negative exponential random variable, and must assume that = (that is m+ ) so that there is no atom of probability at t =, as equivalence would then not be possible. Theorem 4. A Ph (; T ) random variable where T is irreducible and has distinct eigenvalues, is negative exponential with parameter >, (where is the eigenvalue of T of maximal real part) if and only if for all i 2 r i = or l i = : Proof: If e T t = e t then we have from equation (4) i= e it r i l i = e t : (5) For this to be true for all t 2 [; ), it must be that the eigenvalue of T of maximal real part = by the asymptotic behaviour of PH-distributions Neuts (98). Now for equation (5) to be true for all t 2 [; ) also requires that r i l i =, for all i 2, since e it > and i. A necessary and sucient condition for this to be true is that either r i = or l i = for i 2. This is always true if is the left or is the right eigenvector of T corresponding to, but this is not necessary. 4.2 A PH-RENEWAL PROCESS (; T ). For the special case of a Ph-renewal process with Q = T and Q = T, for all k ; t k 2 [; ), e Q t Q : : : Q e Q t k Q e Q t Q : : : Q e Q t k = ( e T t T ) ( e T t 2 T ) : : : e T t k T ( e T t T ) ( e T t 2T ) : : : e T t k = 8 >< >: e T t T e T t ; for k = : e T t k T e T t k ; for k > : (6)
8 Corollary 4.2 A stationary Ph-renewal process (; T ), where T is irreducible and has distinct eigenvalues, is a Poisson process of rate >, (where is the eigenvalue of T of maximal real part) if and only if for all i 2 r i = or l i = : Proof: >From equations (3), (4) and (6) we see that for a stationary Ph-renewal process to be Poisson of rate p p i= i= i e it r i l i e it r i l i = for all t 2 [; ) and p 2 (; 2) ; (7) where = and 2 =. Note that we are considering the stationary process and therefore the initial distribution =, the stationary distribution admitted by the Ph-renewal process. In the case where p =, the proof is basically the same for that of the Ph-random variable and we only consider the case when p =. In this case we write equation (6) as T e T t e T t for all t 2 [; ) ; (8) by noting that T and e T t commute. The next step is to establish a relationship between the stationary distribution and the renewal probability vector. We start by considering from which we see that Q = (T T ) = ; T = ( T ) : >From the assumption of irreducibility we know that T is non-singular and so we can write = ( T ) T : Then by substituting into equation (8), noticing that ( T ) is a scalar quantity and equating to we rearrange to get e T t T e T t = for all t 2 [; ) : Using the spectral form for the inverse of the matrix T, we get P m i= e it r i l i P m i= e it i r i l i = for all t 2 [; ) ;
9 which implies i= ( i + )e it r i l i = for all t 2 [; ): Now, r i and l i are positive since T is a generator matrix (see Seneta (98)), so r l >. Therefore = and r i l i =, for i 2. The result now follows by similar argument to that of Theorem 4:. 4.3 GENERAL MAP s. Theorem 4.3 A stationary general MAP, where Q is irreducible and has distinct eigenvalues, is Poisson of rate >, (where is the eigenvalue of Q of maximal real part) if and only if for all i 2 and for r i = (9) or l i = (2) I R I L def = fi : r i 6= g and (2) def = fi : l i 6= g; (22) we have l i Q r j = ; for all (i; j) 2 I R fi L nfgg: (23) Proof: >From equations (3) and (4) we see that for such a MAP to be Poisson of rate >, we have to satisfy the following i= i= e it r i l i Q : : : e it r i l i Q : : : l= l= l e lt k r l l l e lt k r l l l = ; for all k ; t k 2 [; ): Since r and l are strictly positive (see Seneta (98)), then r l >, so that if we choose k = in (24), we get the following necessary condition (24) = and either r i = or l i = for all i 2: (25) Choosing k = 2 we look at equation (24) again which for all t ; t 2 2 [; ) yields i= i= e it r i l i Q e it r i l i X m j=2 m j=2 j e jt 2 r j l j + e t 2 r l A j e jt 2 r j l j + e t 2 r l A =
10 This in turn gives us another necessary condition r i l i Q r j l j = ; for j 6= ; for all i; (26) which by (2) and (22) reduces to the condition that l i Q r j = ; for (i; j) 2 I R fi L =fgg: (27) We now show that (25) and (27) are also sucient by substituting them into equation (24) which gives us X X e it r i l i Q : : : e jtk r j l j e tk r l i2i X R j2i X R e it r i l i Q : : : e jtk r j l j e tk r l i2i R j2i R = = ; for all k ; t k 2 [; ): (28) Note that if or are left or right eigenvectors respectively corresponding to then all of the above conditions are again trivially satised. Acknowledgements. The authors would like to thank Soren Asmussen for the argument that led to Theorem 4:. We would also like to acknowledge the nancial support of the Australian Research Council through grant A References [] Burke, P.J. (956). The output of a queueing system, Operations Research,4:35{ 65. [2] Neuts, M. (98). Matrix-geometric solutions in stochastic models, The John Hopkins University Press, Baltimore, MD. [3] Olivier, C. and Walrand, J. (994). On the existence of nite-dimensional lters for Markov-modulated trac, Journal of Applied Probability,3:55{525. [4] Seneta, E. (98). Non-negative matrices and Markov chains, Springer-Verlag, New York, Heidelberg, Berlin. [5] Walrand, J. (988). An introduction to queueing networks, Prentice-Hall, Englewood Clis, N.J.
Departure processes from MAP/PH/1 queues
Departure processes from MAP/PH/1 queues David Anthony Green Thesis submitted for the degree of Doctor of Philosophy in Applied Mathematics at The University of Adelaide (Faculty of Mathematical and Computer
More informationDeparture Processes of a Tandem Network
The 7th International Symposium on perations Research and Its Applications (ISRA 08) Lijiang, China, ctober 31 Novemver 3, 2008 Copyright 2008 RSC & APRC, pp. 98 103 Departure Processes of a Tandem Network
More informationModelling Complex Queuing Situations with Markov Processes
Modelling Complex Queuing Situations with Markov Processes Jason Randal Thorne, School of IT, Charles Sturt Uni, NSW 2795, Australia Abstract This article comments upon some new developments in the field
More informationThe Transition Probability Function P ij (t)
The Transition Probability Function P ij (t) Consider a continuous time Markov chain {X(t), t 0}. We are interested in the probability that in t time units the process will be in state j, given that it
More informationIntroduction to Queuing Networks Solutions to Problem Sheet 3
Introduction to Queuing Networks Solutions to Problem Sheet 3 1. (a) The state space is the whole numbers {, 1, 2,...}. The transition rates are q i,i+1 λ for all i and q i, for all i 1 since, when a bus
More informationCHUN-HUA GUO. Key words. matrix equations, minimal nonnegative solution, Markov chains, cyclic reduction, iterative methods, convergence rate
CONVERGENCE ANALYSIS OF THE LATOUCHE-RAMASWAMI ALGORITHM FOR NULL RECURRENT QUASI-BIRTH-DEATH PROCESSES CHUN-HUA GUO Abstract The minimal nonnegative solution G of the matrix equation G = A 0 + A 1 G +
More informationStatistics 150: Spring 2007
Statistics 150: Spring 2007 April 23, 2008 0-1 1 Limiting Probabilities If the discrete-time Markov chain with transition probabilities p ij is irreducible and positive recurrent; then the limiting probabilities
More informationStatistics 992 Continuous-time Markov Chains Spring 2004
Summary Continuous-time finite-state-space Markov chains are stochastic processes that are widely used to model the process of nucleotide substitution. This chapter aims to present much of the mathematics
More informationOnline Companion for. Decentralized Adaptive Flow Control of High Speed Connectionless Data Networks
Online Companion for Decentralized Adaptive Flow Control of High Speed Connectionless Data Networks Operations Research Vol 47, No 6 November-December 1999 Felisa J Vásquez-Abad Départment d informatique
More information2 THE COMPLEXITY OF TORSION-FREENESS On the other hand, the nite presentation of a group G also does not allow us to determine almost any conceivable
THE COMPUTATIONAL COMPLEXITY OF TORSION-FREENESS OF FINITELY PRESENTED GROUPS Steffen Lempp Department of Mathematics University of Wisconsin Madison, WI 53706{1388, USA Abstract. We determine the complexity
More informationMARKOV CHAINS: STATIONARY DISTRIBUTIONS AND FUNCTIONS ON STATE SPACES. Contents
MARKOV CHAINS: STATIONARY DISTRIBUTIONS AND FUNCTIONS ON STATE SPACES JAMES READY Abstract. In this paper, we rst introduce the concepts of Markov Chains and their stationary distributions. We then discuss
More informationOverload Analysis of the PH/PH/1/K Queue and the Queue of M/G/1/K Type with Very Large K
Overload Analysis of the PH/PH/1/K Queue and the Queue of M/G/1/K Type with Very Large K Attahiru Sule Alfa Department of Mechanical and Industrial Engineering University of Manitoba Winnipeg, Manitoba
More informationStochastic Processes
Stochastic Processes 8.445 MIT, fall 20 Mid Term Exam Solutions October 27, 20 Your Name: Alberto De Sole Exercise Max Grade Grade 5 5 2 5 5 3 5 5 4 5 5 5 5 5 6 5 5 Total 30 30 Problem :. True / False
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 informationIEOR 6711, HMWK 5, Professor Sigman
IEOR 6711, HMWK 5, Professor Sigman 1. Semi-Markov processes: Consider an irreducible positive recurrent discrete-time Markov chain {X n } with transition matrix P (P i,j ), i, j S, and finite state space.
More informationRecap. Probability, stochastic processes, Markov chains. ELEC-C7210 Modeling and analysis of communication networks
Recap Probability, stochastic processes, Markov chains ELEC-C7210 Modeling and analysis of communication networks 1 Recap: Probability theory important distributions Discrete distributions Geometric distribution
More informationThe purpose of this paper is to demonstrate the exibility of the SPA
TIPP and the Spectral Expansion Method I. Mitrani 1, A. Ost 2, and M. Rettelbach 2 1 University of Newcastle upon Tyne 2 University of Erlangen-Nurnberg Summary. Stochastic Process Algebras (SPA) like
More informationMidterm 1. Every element of the set of functions is continuous
Econ 200 Mathematics for Economists Midterm Question.- Consider the set of functions F C(0, ) dened by { } F = f C(0, ) f(x) = ax b, a A R and b B R That is, F is a subset of the set of continuous functions
More informationMATH 326: RINGS AND MODULES STEFAN GILLE
MATH 326: RINGS AND MODULES STEFAN GILLE 1 2 STEFAN GILLE 1. Rings We recall first the definition of a group. 1.1. Definition. Let G be a non empty set. The set G is called a group if there is a map called
More informationLinear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) 1.1 The Formal Denition of a Vector Space
Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) Contents 1 Vector Spaces 1 1.1 The Formal Denition of a Vector Space.................................. 1 1.2 Subspaces...................................................
More informationSMSTC (2007/08) Probability.
SMSTC (27/8) Probability www.smstc.ac.uk Contents 12 Markov chains in continuous time 12 1 12.1 Markov property and the Kolmogorov equations.................... 12 2 12.1.1 Finite state space.................................
More informationMarkov Processes Cont d. Kolmogorov Differential Equations
Markov Processes Cont d Kolmogorov Differential Equations The Kolmogorov Differential Equations characterize the transition functions {P ij (t)} of a Markov process. The time-dependent behavior of the
More informationMarkov processes Course note 2. Martingale problems, recurrence properties of discrete time chains.
Institute for Applied Mathematics WS17/18 Massimiliano Gubinelli Markov processes Course note 2. Martingale problems, recurrence properties of discrete time chains. [version 1, 2017.11.1] We introduce
More informationQUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE. Denitizable operators in Krein spaces have spectral properties similar to those
QUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE BRANKO CURGUS and BRANKO NAJMAN Denitizable operators in Krein spaces have spectral properties similar to those of selfadjoint operators in Hilbert spaces.
More informationStability, Queue Length and Delay of Deterministic and Stochastic Queueing Networks Cheng-Shang Chang IBM Research Division T.J. Watson Research Cente
Stability, Queue Length and Delay of Deterministic and Stochastic Queueing Networks Cheng-Shang Chang IBM Research Division T.J. Watson Research Center P.O. Box 704 Yorktown Heights, NY 10598 cschang@watson.ibm.com
More information2 Section 2 However, in order to apply the above idea, we will need to allow non standard intervals ('; ) in the proof. More precisely, ' and may gene
Introduction 1 A dierential intermediate value theorem by Joris van der Hoeven D pt. de Math matiques (B t. 425) Universit Paris-Sud 91405 Orsay Cedex France June 2000 Abstract Let T be the eld of grid-based
More informationVector Space Basics. 1 Abstract Vector Spaces. 1. (commutativity of vector addition) u + v = v + u. 2. (associativity of vector addition)
Vector Space Basics (Remark: these notes are highly formal and may be a useful reference to some students however I am also posting Ray Heitmann's notes to Canvas for students interested in a direct computational
More informationUniversity of California. Berkeley, CA fzhangjun johans lygeros Abstract
Dynamical Systems Revisited: Hybrid Systems with Zeno Executions Jun Zhang, Karl Henrik Johansson y, John Lygeros, and Shankar Sastry Department of Electrical Engineering and Computer Sciences University
More informationEigenvalue comparisons in graph theory
Eigenvalue comparisons in graph theory Gregory T. Quenell July 1994 1 Introduction A standard technique for estimating the eigenvalues of the Laplacian on a compact Riemannian manifold M with bounded curvature
More informationG-networks with synchronized partial ushing. PRi SM, Universite de Versailles, 45 av. des Etats Unis, Versailles Cedex,France
G-networks with synchronized partial ushing Jean-Michel FOURNEAU ;a, Dominique VERCH ERE a;b a PRi SM, Universite de Versailles, 45 av. des Etats Unis, 78 05 Versailles Cedex,France b CERMSEM, Universite
More informationAdvanced Queueing Theory
Advanced Queueing Theory 1 Networks of queues (reversibility, output theorem, tandem networks, partial balance, product-form distribution, blocking, insensitivity, BCMP networks, mean-value analysis, Norton's
More informationLinear Algebra Practice Problems
Linear Algebra Practice Problems Math 24 Calculus III Summer 25, Session II. Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated,
More informationonly nite eigenvalues. This is an extension of earlier results from [2]. Then we concentrate on the Riccati equation appearing in H 2 and linear quadr
The discrete algebraic Riccati equation and linear matrix inequality nton. Stoorvogel y Department of Mathematics and Computing Science Eindhoven Univ. of Technology P.O. ox 53, 56 M Eindhoven The Netherlands
More informationMulti Stage Queuing Model in Level Dependent Quasi Birth Death Process
International Journal of Statistics and Systems ISSN 973-2675 Volume 12, Number 2 (217, pp. 293-31 Research India Publications http://www.ripublication.com Multi Stage Queuing Model in Level Dependent
More information(b) What is the variance of the time until the second customer arrives, starting empty, assuming that we measure time in minutes?
IEOR 3106: Introduction to Operations Research: Stochastic Models Fall 2006, Professor Whitt SOLUTIONS to Final Exam Chapters 4-7 and 10 in Ross, Tuesday, December 19, 4:10pm-7:00pm Open Book: but only
More informationExperimental evidence showing that stochastic subspace identication methods may fail 1
Systems & Control Letters 34 (1998) 303 312 Experimental evidence showing that stochastic subspace identication methods may fail 1 Anders Dahlen, Anders Lindquist, Jorge Mari Division of Optimization and
More informationMarkov Chains, Stochastic Processes, and Matrix Decompositions
Markov Chains, Stochastic Processes, and Matrix Decompositions 5 May 2014 Outline 1 Markov Chains Outline 1 Markov Chains 2 Introduction Perron-Frobenius Matrix Decompositions and Markov Chains Spectral
More information[4] T. I. Seidman, \\First Come First Serve" is Unstable!," tech. rep., University of Maryland Baltimore County, 1993.
[2] C. J. Chase and P. J. Ramadge, \On real-time scheduling policies for exible manufacturing systems," IEEE Trans. Automat. Control, vol. AC-37, pp. 491{496, April 1992. [3] S. H. Lu and P. R. Kumar,
More informationgrowth rates of perturbed time-varying linear systems, [14]. For this setup it is also necessary to study discrete-time systems with a transition map
Remarks on universal nonsingular controls for discrete-time systems Eduardo D. Sontag a and Fabian R. Wirth b;1 a Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, b sontag@hilbert.rutgers.edu
More informationMAT SYS 5120 (Winter 2012) Assignment 5 (not to be submitted) There are 4 questions.
MAT 4371 - SYS 5120 (Winter 2012) Assignment 5 (not to be submitted) There are 4 questions. Question 1: Consider the following generator for a continuous time Markov chain. 4 1 3 Q = 2 5 3 5 2 7 (a) Give
More informationPerron Frobenius Theory
Perron Frobenius Theory Oskar Perron Georg Frobenius (1880 1975) (1849 1917) Stefan Güttel Perron Frobenius Theory 1 / 10 Positive and Nonnegative Matrices Let A, B R m n. A B if a ij b ij i, j, A > B
More informationAnalysis on Graphs. Alexander Grigoryan Lecture Notes. University of Bielefeld, WS 2011/12
Analysis on Graphs Alexander Grigoryan Lecture Notes University of Bielefeld, WS 0/ Contents The Laplace operator on graphs 5. The notion of a graph............................. 5. Cayley graphs..................................
More informationConvergence Rates for Renewal Sequences
Convergence Rates for Renewal Sequences M. C. Spruill School of Mathematics Georgia Institute of Technology Atlanta, Ga. USA January 2002 ABSTRACT The precise rate of geometric convergence of nonhomogeneous
More informationFast Estimation of the Statistics of Excessive Backlogs in Tandem Networks of Queues.
Fast Estimation of the Statistics of Excessive Backlogs in Tandem Networks of Queues. M.R. FRATER and B.D.O. ANDERSON Department of Systems Engineering Australian National University The estimation of
More informationOPTIMALITY OF RANDOMIZED TRUNK RESERVATION FOR A PROBLEM WITH MULTIPLE CONSTRAINTS
OPTIMALITY OF RANDOMIZED TRUNK RESERVATION FOR A PROBLEM WITH MULTIPLE CONSTRAINTS Xiaofei Fan-Orzechowski Department of Applied Mathematics and Statistics State University of New York at Stony Brook Stony
More informationModel reversibility of a two dimensional reflecting random walk and its application to queueing network
arxiv:1312.2746v2 [math.pr] 11 Dec 2013 Model reversibility of a two dimensional reflecting random walk and its application to queueing network Masahiro Kobayashi, Masakiyo Miyazawa and Hiroshi Shimizu
More informationLinear Algebra: Linear Systems and Matrices - Quadratic Forms and Deniteness - Eigenvalues and Markov Chains
Linear Algebra: Linear Systems and Matrices - Quadratic Forms and Deniteness - Eigenvalues and Markov Chains Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 3, 3 Systems
More informationCS 798: Homework Assignment 3 (Queueing Theory)
1.0 Little s law Assigned: October 6, 009 Patients arriving to the emergency room at the Grand River Hospital have a mean waiting time of three hours. It has been found that, averaged over the period of
More informationChapter 30 Minimality and Stability of Interconnected Systems 30.1 Introduction: Relating I/O and State-Space Properties We have already seen in Chapt
Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology 1 1 c Chapter
More informationChapter 5. Continuous-Time Markov Chains. Prof. Shun-Ren Yang Department of Computer Science, National Tsing Hua University, Taiwan
Chapter 5. Continuous-Time Markov Chains Prof. Shun-Ren Yang Department of Computer Science, National Tsing Hua University, Taiwan Continuous-Time Markov Chains Consider a continuous-time stochastic process
More informationCover Page. The handle holds various files of this Leiden University dissertation
Cover Page The handle http://hdlhandlenet/1887/39637 holds various files of this Leiden University dissertation Author: Smit, Laurens Title: Steady-state analysis of large scale systems : the successive
More informationSTABILITY OF INVARIANT SUBSPACES OF COMMUTING MATRICES We obtain some further results for pairs of commuting matrices. We show that a pair of commutin
On the stability of invariant subspaces of commuting matrices Tomaz Kosir and Bor Plestenjak September 18, 001 Abstract We study the stability of (joint) invariant subspaces of a nite set of commuting
More informationIEOR 6711: Stochastic Models I Professor Whitt, Thursday, November 29, Weirdness in CTMC s
IEOR 6711: Stochastic Models I Professor Whitt, Thursday, November 29, 2012 Weirdness in CTMC s Where s your will to be weird? Jim Morrison, The Doors We are all a little weird. And life is a little weird.
More informationRegenerative Processes. Maria Vlasiou. June 25, 2018
Regenerative Processes Maria Vlasiou June 25, 218 arxiv:144.563v1 [math.pr] 22 Apr 214 Abstract We review the theory of regenerative processes, which are processes that can be intuitively seen as comprising
More informationContents. 6 Systems of First-Order Linear Dierential Equations. 6.1 General Theory of (First-Order) Linear Systems
Dierential Equations (part 3): Systems of First-Order Dierential Equations (by Evan Dummit, 26, v 2) Contents 6 Systems of First-Order Linear Dierential Equations 6 General Theory of (First-Order) Linear
More informationNon-Essential Uses of Probability in Analysis Part IV Efficient Markovian Couplings. Krzysztof Burdzy University of Washington
Non-Essential Uses of Probability in Analysis Part IV Efficient Markovian Couplings Krzysztof Burdzy University of Washington 1 Review See B and Kendall (2000) for more details. See also the unpublished
More informationSpurious Chaotic Solutions of Dierential. Equations. Sigitas Keras. September Department of Applied Mathematics and Theoretical Physics
UNIVERSITY OF CAMBRIDGE Numerical Analysis Reports Spurious Chaotic Solutions of Dierential Equations Sigitas Keras DAMTP 994/NA6 September 994 Department of Applied Mathematics and Theoretical Physics
More informationECON Answers Homework #2
ECON 33 - Answers Homework #2 Exercise : Denote by x the number of containers of tye H roduced, y the number of containers of tye T and z the number of containers of tye I. There are 3 inut equations that
More informationMarkov Chains and Stochastic Sampling
Part I Markov Chains and Stochastic Sampling 1 Markov Chains and Random Walks on Graphs 1.1 Structure of Finite Markov Chains We shall only consider Markov chains with a finite, but usually very large,
More informationOn the Pathwise Optimal Bernoulli Routing Policy for Homogeneous Parallel Servers
On the Pathwise Optimal Bernoulli Routing Policy for Homogeneous Parallel Servers Ger Koole INRIA Sophia Antipolis B.P. 93, 06902 Sophia Antipolis Cedex France Mathematics of Operations Research 21:469
More information2905 Queueing Theory and Simulation PART III: HIGHER DIMENSIONAL AND NON-MARKOVIAN QUEUES
295 Queueing Theory and Simulation PART III: HIGHER DIMENSIONAL AND NON-MARKOVIAN QUEUES 16 Queueing Systems with Two Types of Customers In this section, we discuss queueing systems with two types of customers.
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationMarkov chains. 1 Discrete time Markov chains. c A. J. Ganesh, University of Bristol, 2015
Markov chains c A. J. Ganesh, University of Bristol, 2015 1 Discrete time Markov chains Example: A drunkard is walking home from the pub. There are n lampposts between the pub and his home, at each of
More information8 Extremal Values & Higher Derivatives
8 Extremal Values & Higher Derivatives Not covered in 2016\17 81 Higher Partial Derivatives For functions of one variable there is a well-known test for the nature of a critical point given by the sign
More informationMATH37012 Week 10. Dr Jonathan Bagley. Semester
MATH37012 Week 10 Dr Jonathan Bagley Semester 2-2018 2.18 a) Finding and µ j for a particular category of B.D. processes. Consider a process where the destination of the next transition is determined by
More informationWhat is A + B? What is A B? What is AB? What is BA? What is A 2? and B = QUESTION 2. What is the reduced row echelon matrix of A =
STUDENT S COMPANIONS IN BASIC MATH: THE ELEVENTH Matrix Reloaded by Block Buster Presumably you know the first part of matrix story, including its basic operations (addition and multiplication) and row
More information8. Statistical Equilibrium and Classification of States: Discrete Time Markov Chains
8. Statistical Equilibrium and Classification of States: Discrete Time Markov Chains 8.1 Review 8.2 Statistical Equilibrium 8.3 Two-State Markov Chain 8.4 Existence of P ( ) 8.5 Classification of States
More information2. Prime and Maximal Ideals
18 Andreas Gathmann 2. Prime and Maximal Ideals There are two special kinds of ideals that are of particular importance, both algebraically and geometrically: the so-called prime and maximal ideals. Let
More informationform, but that fails as soon as one has an object greater than every natural number. Induction in the < form frequently goes under the fancy name \tra
Transnite Ordinals and Their Notations: For The Uninitiated Version 1.1 Hilbert Levitz Department of Computer Science Florida State University levitz@cs.fsu.edu Intoduction This is supposed to be a primer
More informationNew concepts: Span of a vector set, matrix column space (range) Linearly dependent set of vectors Matrix null space
Lesson 6: Linear independence, matrix column space and null space New concepts: Span of a vector set, matrix column space (range) Linearly dependent set of vectors Matrix null space Two linear systems:
More informationX. Hu, R. Shonkwiler, and M.C. Spruill. School of Mathematics. Georgia Institute of Technology. Atlanta, GA 30332
Approximate Speedup by Independent Identical Processing. Hu, R. Shonkwiler, and M.C. Spruill School of Mathematics Georgia Institute of Technology Atlanta, GA 30332 Running head: Parallel iip Methods Mail
More informationLecture 10: Semi-Markov Type Processes
Lecture 1: Semi-Markov Type Processes 1. Semi-Markov processes (SMP) 1.1 Definition of SMP 1.2 Transition probabilities for SMP 1.3 Hitting times and semi-markov renewal equations 2. Processes with semi-markov
More informationTHE VARIANCE CONSTANT FOR THE ACTUAL WAITING TIME OF THE PH/PH/1 QUEUE. By Mogens Bladt National University of Mexico
The Annals of Applied Probability 1996, Vol. 6, No. 3, 766 777 THE VARIANCE CONSTANT FOR THE ACTUAL WAITING TIME OF THE PH/PH/1 QUEUE By Mogens Bladt National University of Mexico In this paper we consider
More informationQueueing systems in a random environment with applications
Queueing systems in a random environment with applications Ruslan Krenzler, Hans Daduna Universität Hamburg OR 2013 3.-6. September 2013 Krenzler, Daduna (Uni HH) Queues in rnd. environment OR 2013 1 /
More informationIrreducibility. Irreducible. every state can be reached from every other state For any i,j, exist an m 0, such that. Absorbing state: p jj =1
Irreducibility Irreducible every state can be reached from every other state For any i,j, exist an m 0, such that i,j are communicate, if the above condition is valid Irreducible: all states are communicate
More informationSTA 624 Practice Exam 2 Applied Stochastic Processes Spring, 2008
Name STA 624 Practice Exam 2 Applied Stochastic Processes Spring, 2008 There are five questions on this test. DO use calculators if you need them. And then a miracle occurs is not a valid answer. There
More informationLecture 20: Reversible Processes and Queues
Lecture 20: Reversible Processes and Queues 1 Examples of reversible processes 11 Birth-death processes We define two non-negative sequences birth and death rates denoted by {λ n : n N 0 } and {µ n : n
More information4.1 Eigenvalues, Eigenvectors, and The Characteristic Polynomial
Linear Algebra (part 4): Eigenvalues, Diagonalization, and the Jordan Form (by Evan Dummit, 27, v ) Contents 4 Eigenvalues, Diagonalization, and the Jordan Canonical Form 4 Eigenvalues, Eigenvectors, and
More informationA TANDEM QUEUE WITH SERVER SLOW-DOWN AND BLOCKING
Stochastic Models, 21:695 724, 2005 Copyright Taylor & Francis, Inc. ISSN: 1532-6349 print/1532-4214 online DOI: 10.1081/STM-200056037 A TANDEM QUEUE WITH SERVER SLOW-DOWN AND BLOCKING N. D. van Foreest
More informationBalance properties of multi-dimensional words
Theoretical Computer Science 273 (2002) 197 224 www.elsevier.com/locate/tcs Balance properties of multi-dimensional words Valerie Berthe a;, Robert Tijdeman b a Institut de Mathematiques de Luminy, CNRS-UPR
More informationSOME MEASURABILITY AND CONTINUITY PROPERTIES OF ARBITRARY REAL FUNCTIONS
LE MATEMATICHE Vol. LVII (2002) Fasc. I, pp. 6382 SOME MEASURABILITY AND CONTINUITY PROPERTIES OF ARBITRARY REAL FUNCTIONS VITTORINO PATA - ALFONSO VILLANI Given an arbitrary real function f, the set D
More informationOne important issue in the study of queueing systems is to characterize departure processes. Study on departure processes was rst initiated by Burke (
The Departure Process of the GI/G/ Queue and Its MacLaurin Series Jian-Qiang Hu Department of Manufacturing Engineering Boston University 5 St. Mary's Street Brookline, MA 2446 Email: hqiang@bu.edu June
More informationOperations Research Letters. Instability of FIFO in a simple queueing system with arbitrarily low loads
Operations Research Letters 37 (2009) 312 316 Contents lists available at ScienceDirect Operations Research Letters journal homepage: www.elsevier.com/locate/orl Instability of FIFO in a simple queueing
More information1 IEOR 4701: Continuous-Time Markov Chains
Copyright c 2006 by Karl Sigman 1 IEOR 4701: Continuous-Time Markov Chains A Markov chain in discrete time, {X n : n 0}, remains in any state for exactly one unit of time before making a transition (change
More informationLECTURE NOTES: Discrete time Markov Chains (2/24/03; SG)
LECTURE NOTES: Discrete time Markov Chains (2/24/03; SG) A discrete time Markov Chains is a stochastic dynamical system in which the probability of arriving in a particular state at a particular time moment
More informationChapter 1. Introduction. 1.1 Stochastic process
Chapter 1 Introduction Process is a phenomenon that takes place in time. In many practical situations, the result of a process at any time may not be certain. Such a process is called a stochastic process.
More informationExact Simulation of the Stationary Distribution of M/G/c Queues
1/36 Exact Simulation of the Stationary Distribution of M/G/c Queues Professor Karl Sigman Columbia University New York City USA Conference in Honor of Søren Asmussen Monday, August 1, 2011 Sandbjerg Estate
More informationConvex Optimization CMU-10725
Convex Optimization CMU-10725 Simulated Annealing Barnabás Póczos & Ryan Tibshirani Andrey Markov Markov Chains 2 Markov Chains Markov chain: Homogen Markov chain: 3 Markov Chains Assume that the state
More informationMath Camp Notes: Linear Algebra II
Math Camp Notes: Linear Algebra II Eigenvalues Let A be a square matrix. An eigenvalue is a number λ which when subtracted from the diagonal elements of the matrix A creates a singular matrix. In other
More informationAn M/M/1 Queue in Random Environment with Disasters
An M/M/1 Queue in Random Environment with Disasters Noam Paz 1 and Uri Yechiali 1,2 1 Department of Statistics and Operations Research School of Mathematical Sciences Tel Aviv University, Tel Aviv 69978,
More informationPerturbation results for nearly uncoupled Markov. chains with applications to iterative methods. Jesse L. Barlow. December 9, 1992.
Perturbation results for nearly uncoupled Markov chains with applications to iterative methods Jesse L. Barlow December 9, 992 Abstract The standard perturbation theory for linear equations states that
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 information290 J.M. Carnicer, J.M. Pe~na basis (u 1 ; : : : ; u n ) consisting of minimally supported elements, yet also has a basis (v 1 ; : : : ; v n ) which f
Numer. Math. 67: 289{301 (1994) Numerische Mathematik c Springer-Verlag 1994 Electronic Edition Least supported bases and local linear independence J.M. Carnicer, J.M. Pe~na? Departamento de Matematica
More informationAdaptive linear quadratic control using policy. iteration. Steven J. Bradtke. University of Massachusetts.
Adaptive linear quadratic control using policy iteration Steven J. Bradtke Computer Science Department University of Massachusetts Amherst, MA 01003 bradtke@cs.umass.edu B. Erik Ydstie Department of Chemical
More informationCentral limit theorems for ergodic continuous-time Markov chains with applications to single birth processes
Front. Math. China 215, 1(4): 933 947 DOI 1.17/s11464-15-488-5 Central limit theorems for ergodic continuous-time Markov chains with applications to single birth processes Yuanyuan LIU 1, Yuhui ZHANG 2
More informationLecture 7. µ(x)f(x). When µ is a probability measure, we say µ is a stationary distribution.
Lecture 7 1 Stationary measures of a Markov chain We now study the long time behavior of a Markov Chain: in particular, the existence and uniqueness of stationary measures, and the convergence of the distribution
More informationGeometric ρ-mixing property of the interarrival times of a stationary Markovian Arrival Process
Author manuscript, published in "Journal of Applied Probability 50, 2 (2013) 598-601" Geometric ρ-mixing property of the interarrival times of a stationary Markovian Arrival Process L. Hervé and J. Ledoux
More informationreversed chain is ergodic and has the same equilibrium probabilities (check that π j =
Lecture 10 Networks of queues In this lecture we shall finally get around to consider what happens when queues are part of networks (which, after all, is the topic of the course). Firstly we shall need
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 informationSystem theory and system identification of compartmental systems Hof, Jacoba Marchiena van den
University of Groningen System theory and system identification of compartmental systems Hof, Jacoba Marchiena van den IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF)
More information