We develop anapproximation for the buffer overflow probability of a stable tandem network in dimensions three or more. The overflow event in terms of the constrained random walk representing the network is the following: the sum of the components of the process hits n before hitting 0. This is one of the most commonly studied rare events in the context of queueing systems and the constrained processes representing them. The approximation is valid for almost all initial points of the process and its relative error decays exponentially in n. The analysis is based on an affine transformation of the process and the problem; as $n\rightarrow \infty$ the transformed process converges to an unstable constrained random walk. The approximation formula consists of the probability of the limit unstable process hitting a limit boundary in finite time. We give an explicit formula for this probability in terms of the utilization rates of the nodes of the network.

The classical optimal trading problem is the closure of a position in an asset over a time interval; the trader maximizes the expected revenues under the constraint that the position be closed by terminal time. Since the asset price is stochastic, the liquidation constraint is too restrictive; the trader may want to relax it or slow down/stop trading depending on price behavior. We consider two additional parameters that serve these purposes within the Almgren-Chriss framework: a binary valued process that prescribes when trading takes place and a set that prescribes when full liquidation is required. The permanent price impact parameter enters the problem as the negative part of the terminal cost. A terminal cost that can take negative values implies that the BSDE associated with the value function of the control problem can explode backward in time and that existence results on solutions of BSDE with singular terminal values are not directly applicable. When liquidation costs are quadratic, the problem is convex and, under a general filtration, the minimal supersolution of the BSDE gives the value function and the optimal control. For the non-quadratic case, we assume a stochastic Markovian volatility model. These give PDE/PDE-system representations for the value functions.

We study a class of nonlinear BSDEs with a superlinear driver process f adapted to a filtration F and over a random time interval [[0, S]] where S is a stopping time of F. The terminal condition $\xi$ is allowed to take the value +$\infty$, i.e., singular. Our goal is to show existence of solutions to the BSDE in this setting. We will do so by proving that the minimal supersolution to the BSDE is a solution, i.e., attains the terminal values with probability 1. We consider three types of terminal values: 1) Markovian: i.e., $\xi$ is of the form $\xi$ = g($\Xi$ S) where $\Xi$ is a continuous Markovian diffusion process and S is a hitting time of $\Xi$ and g is a deterministic function 2) terminal conditions of the form $\xi$ = $\infty$ $\times$ 1 {$\tau$ $\le$S} and 3) $\xi$ 2 = $\infty$ $\times$ 1 {$\tau$ >S} where $\tau$ is another stopping time. For general $\xi$ we prove the minimal supersolution is continuous at time S provided that F is left continuous at time S. We call a stopping time S solvable with respect to a given BSDE and filtration if the BSDE has a minimal supersolution with terminal value $\infty$ at terminal time S. The concept of solvability plays a key role in many of the arguments. Finally, we discuss implications of our results on the Markovian terminal conditions to solution of nonlinear elliptic PDE with singular boundary conditions.

Let X be the constrained random walk on $${\mathbb {Z}}_+^2$$ having increments (1, 0), $$(-\,1,1)$$ , $$(0,-\,1)$$ with jump probabilities $$\lambda (M_k)$$ , $$\mu _1(M_k)$$ , and $$\mu _2(M_k)$$ where M is an irreducible aperiodic finite state Markov chain. The process X represents the lengths of two tandem queues with arrival rate $$\lambda (M_k)$$ , and service rates $$\mu _1(M_k)$$ , and $$\mu _2(M_k)$$ ; the process M represents the random environment within which the system operates. We assume that the average arrival rate with respect to the stationary measure of M is less than the average service rates, i.e., X is assumed stable. Let $$\tau _n$$ be the first time when the sum of the components of X equals n for the first time. Let Y be the random walk on $${{\mathbb {Z}}} \times {{\mathbb {Z}}}_+$$ having increments $$(-\,1,0)$$ , (1, 1), $$(0,-\,1)$$ with probabilities $$\lambda (M_k)$$ , $$\mu _1(M_k)$$ , and $$\mu _2(M_k)$$ . Supposing that the queues share a joint buffer of size n, $$p_n =P_{(x_n,m)}(\tau _n < \tau _0)$$ is the probability that this buffer overflows during a busy cycle of the system. To the best of our knowledge, the only methods currently available for the approximation of $$p_n$$ are classical large deviations analysis giving the exponential decay rate of $$p_n$$ and rare event simulation. Let $$\tau $$ be the first time the components of Y are equal. For $$x \in {{\mathbb {R}}}_+^2$$ , $$x(1) + x(2) < 1$$ , $$x(1) > 0$$ , and $$x_n = \lfloor nx \rfloor $$ , we show that $$P_{(n-x_n(1),x_n(2),m)}( \tau < \infty )$$ approximates $$P_{(x_n,m)}(\tau _n < \tau _0)$$ with exponentially vanishing relative error as $$n\rightarrow \infty $$ . For the analysis we define a characteristic matrix in terms of the jump probabilities of (X, M). The 0-level set of the characteristic polynomial of this matrix defines the characteristic surface; conjugate points on this surface and the associated eigenvectors of the characteristic matrix are used to define (sub/super) harmonic functions which play a fundamental role both in our analysis and the computation/approximation of $$P_{(y,m)}(\tau < \infty )$$ .

Let X be the constrained random walk on $${\mathbb Z}_+^2$$ with increments (1, 0), $$(-1,0)$$ , (0, 1) and $$(0,-1)$$ ; X represents, at arrivals and service completions, the lengths of two queues (or two stacks in computer science applications) working in parallel whose service and interarrival times are exponentially distributed with arrival rates $$\lambda _i$$ and service rates $$\mu _i$$ , $$i=1,2$$ ; we assume $$\lambda _i < \mu _i$$ , $$i=1,2$$ , i.e., X is assumed stable. Without loss of generality we assume $$\rho _1 =\lambda _1/\mu _1 \geqslant \rho _2 = \lambda _2/\mu _2$$ . Let $$\tau _n$$ be the first time X hits the line $$\partial A_n = \{x \in {\mathbb Z}^2:x(1)+x(2) = n \}$$ , i.e., when the sum of the components of X equals n for the first time. Let Y be the same random walk as X but only constrained on $$\{y \in {{\mathbb {Z}}}^2: y(2)=0\}$$ and its jump probabilities for the first component reversed. Let $$\partial B =\{y \in {{\mathbb {Z}}}^2: y(1) = y(2) \}$$ and let $$\tau $$ be the first time Y hits $$\partial B$$ . The probability $$p_n = P_x(\tau _n < \tau _0)$$ is a key performance measure of the queueing system (or the two stacks) represented by X (if the queues/stacks share a common buffer, then $$p_n$$ is the probability that this buffer overflows during the system’s first busy cycle). Stability of the process implies that $$p_n$$ decays exponentially in n when the process starts off the exit boundary $$\partial A_n.$$ We show that, for $$x_n= \lfloor nx \rfloor $$ , $$x \in {{\mathbb {R}}}_+^2$$ , $$x(1)+x(2) \leqslant 1$$ , $$x(1) > 0$$ , $$P_{(n-x_n(1),x_n(2))}( \tau < \infty )$$ approximates $$P_{x_n}(\tau _n < \tau _0)$$ with exponentially vanishing relative error. Let $$r = (\lambda _1 + \lambda _2)/(\mu _1 + \mu _2)$$ ; for $$r^2 < \rho _2$$ and $$\rho _1 \ne \rho _2$$ , we construct a class of harmonic functions from single and conjugate points on a related characteristic surface for Y with which the probability $$P_y(\tau < \infty )$$ can be approximated with bounded relative error. For $$r^2 = \rho _1 \rho _2$$ , we obtain the exact formula $$P_y(\tau < \infty ) = r^{y(1)-y(2)} +\frac{r(1-r)}{r-\rho _2}\left( \rho _1^{y(1)} - r^{y(1)-y(2)} \rho _1^{y(2)}\right) .$$

Let X be the constrained random walk on Z(+)(2) having increments (1, 0), (-1, 1), and (0, -1) with respective probabilities A lambda,mu 1, and mu 2 representing the lengths of two tandem queues. We assume that X is stable and mu 1 not equal mu 2. Let tau(n) be the first time when the sum of the components of X equals n. Let Y be the constrained random walk on Z x Z(+) having increments (-1, 0), (1, 1), and (0, -1) with probabilities lambda, mu(1), and mu(2). Let tau be the first time that the components of Y are equal to each other. We prove that Pn-xn(1),x(n)(2)(tau < infinity) approximates p(n)(x(n)) with relative error exponentially decaying in n for x(n) = [n(x)], x is an element of R-+(2), 0 < x(1) + x(2) < 1, x (1) > 0. An affine transformation moving the origin to the point (n, 0) and letting n -> infinity connect the X and Y processes. We use a linear combination of basis functions constructed from single and conjugate points on a characteristic surface associated with X to derive a simple expression for P-y (tau < infinity) in terms of the utilization rates of the nodes. The proof that the relative error decays exponentially in n uses a sequence of subsolutions of a related HamiltonJacobi-Bellman equation on a manifold consisting of three copies of R-+(2) glued to each other along the constraining boundaries. We indicate how the ideas of the paper can be generalized to more general processes and other exit boundaries.

For a nite state Markov process and a nite collection f k;k2 Kg of subsets of its state space, let k be the rst time the process visits the set k. We derive explicit/recursive formulas for the joint density and tail probabilities of the family of stopping timesf k;k2 Kg. In particular, we provide a general solution to the problem that was studied (Assaf et. al., Multivariate phase-type distributions, Operations Research 32 (1984), no. 3, 688-702) in the context of multivariate phase-type distributions. We give a numerical example and indicate the relevance of our results to credit risk modeling.

We study a generalization of the $M/G/1$ system (denoted by $rM/G/1$) with independent and identically distributed (iid) service times and with an arrival process whose arrival rate $\lambda_0f(r)$ depends on the remaining service time $r$ of the current customer being served. We derive a natural stability condition and provide a stationary analysis under it both at service completion times (of the queue length process) and in continuous time (of the queue length and the residual service time). In particular, we show that the stationary measure of queue length at service completion times is equal to that of a corresponding $M/G/1$ system. For $f > 0$ we show that the continuous time stationary measure of the $rM/G/1$ system is linked to the $M/G/1$ system via a time change. As opposed to the $M/G/1$ queue, the stationary measure of queue length of the $rM/G/1$ system at service completions differs from its marginal distribution under the continuous time stationary measure. Thus, in general, arrivals of the $rM/G/1$ system do not see time averages. We derive formulas for the average queue length, probability of an empty system and average waiting time under the continuous time stationary measure. We provide examples showing the effect of changing the reshaping function on the average waiting time.

EXIT PROBABILITIES OF MARKOV MODULATED CONSTRAINED RANDOM WALKS Başoğlu Kabran, Fatma Ph.D., Department of Financial Mathematics Supervisor : Assoc. Prof. Dr. Ali Devin Sezer September 2018, 77 pages Let X be the constrained random walk on Z+ with increments (0, 0), (1, 0), (−1, 1), (0,−1) whose jump probabilities are determined by the state of a finite state Markov chain M . X represents the lengths of two queues of customers (or packets, tasks, etc.) waiting for service from two servers working in tandem; the arrival of customers occur with rate λ(Mk), service takes place at rates μ1(Mk), and μ2(Mk) where Mk denotes the current state of the Markov chain M . We assume that the average arrival rate is less than the average service rates, i.e., X is assumed stable. Stability implies that X moves in cycles that restart each time it hits the origin. Let τn be the first time X hits the line ∂An = {x : x(1) + x(2) = n}, i.e., when the sum of the queue lengths equals n for the first time ; if the queues share a common buffer, τn represents the time of a buffer overflow and pn = P(x,m)(τn < τ0) is the probability that a given cycle ends with a buffer overflow, i.e., system failure. Let Y be the same random walk as X but only constrained on ∂2 = {y ∈ Z × Z+ : y(2) = 0} and its jump probabilities for the first component reversed. Let B = {y ∈ Z : y(1) = y(2)} and let τ be the first time Y hits B. For x ∈ R+, with x(1) + x(2) < 1 define xn = ⌊nx⌋ and let m ∈ M denote the initial point of the Markov chain M . We show that P((n−xn(1),xn(2)),m)(τ < ∞) approximates P((xn(1),xn(2)),m)(τn < τ0) with exponentially vanishing relative error when x(1) > 0. We then construct a class of harmonic functions for (Y,M) and use their linear combinations to develop approximate formulas for P(y,m)(τ < ∞). The construction is based on points on a vii characteristic surface associated with Y defined through the eigenvalues of a matrix whose components depend on the transition matrix of the modulating chain and the jump probabilities of Y . We indicate possible applications of our results and approach in finance and insurance.

A classical problem in computer science going back to [4, section 2.2.2, exercise 13], is the analysis of dynamic storage (memory) allocation algorithms. A basic mathematical model used for this analysis is that of a constrained random walk X on the positive orthant Z+ with increments V = {−ei,+ei, i = 1, 2, 3, ..., d}, where {ei, i = 1, 2, 3, ..., d} is the standard basis for R (the increments of the walk are set to 0 when it attempts to leave the orthant), i.e.,

Let $X$ be the constrained random walk on ${\mathbb Z}_+^d$ representing the queue lengths of a stable Jackson network and $x$ its initial position. Let $\tau_n$ be the first time the sum of the components of $X$ equals $n$. $p_n \doteq P_x(\tau_n < \tau_0)$ is a key performance measure for the queueing system represented by $X$, stability implies $p_n\rightarrow 0$ exponentially. Currently the only analytic method available to approximate $p_n$ is large deviations analysis, which gives the exponential decay rate of $p_n$. Finer results are available via rare event simulation. The present article develops a new method to approximate $p_n$ and related expectations. The method has two steps: 1) with an affine transformation, move the origin onto the exit boundary of $\tau_n$, take limits to remove some of the constraints on the dynamics, this yields a limit unstable constrained walk $Y$ 2) Construct a basis of harmonic functions of $Y$ and use them to apply the classical superposition principle of linear analysis. The basis functions are linear combinations of $\log$-linear functions and come from solutions of "harmonic systems," which are graphs whose vertices represent points on the "characteristic surface" of $Y$, the edges between the vertices represent conjugacy relations between the points, the loops represent membership in "the boundary characteristic surfaces." Using our method we derive explicit, simple and almost exact formulas for $P_x(\tau_n < \tau_0)$ for $d$-tandem queues, similar to the product form formulas for the stationary distribution of $X$. The same method allows us to approximate the Balayage operator mapping $f$ to $x \rightarrow {\mathbb E}_x \left[ f(X_{\tau_n}) 1_{\{\tau_n < \tau_0\}} \right]$ for a range of stable constrained random walks in $2$ dimensions. We indicate how the ideas of the paper relate to more general processes and exit boundaries.

Consider a fully connected network of nodes, some of which have a piece of data to be disseminated to the whole network. We analyze the following push-type epidemic algorithm: in each push round, every node that has the data, i.e., every infected node, randomly chooses c E Z. other nodes in the network and transmits, i.e., pushes, the data to them. We write this round as a random walk whose each step corresponds to a random selection of one of the infected nodes; this gives recursive formulas for the distribution and the moments of the number of newly infected nodes in a push round. We use the formula for the distribution to compute the expected number of rounds so that a given percentage of the network is infected and continue a numerical comparison of the push algorithm and the pull algorithm (where the susceptible nodes randomly choose peers) initiated in an earlier work. We then derive the fluid and diffusion limits of the random walk as the network size goes to infinity and deduce a number of properties of the push algorithm: (1) the number of newly infected nodes in a push round, and the number of random selections needed so that a given percent of the network is infected, are both asymptotically normal, (2) for large networks, starting with a nonzero proportion of infected nodes, a pull round infects slightly more nodes on average, (3) the number of rounds until a given proportion lambda of the network is infected converges to a constant for almost all lambda is an element of (0, 1). Numerical examples for theoretical results are provided. (C) 2014 Elsevier ay. All rights reserved.

It is probably fair to say that probability theory (more generally analysis), optimization and the interplay between them form the overarching mathematical themes of mathematical finance. By optimi...

We consider a hypothetical company that is assumed to have just manufactured and sold a number of copies of a product. It is known that, with a small probability, the company has committed a manufacturing fault that will require a recall. The company is able to observe the expiration times of the sold items whose distribution depends on whether the fault is present or absent. At the expiration of each item, a public inspection takes place that may reveal the fault, if it exists. Based on this information, the company can recall the product at any moment and pay back each customer the price of the product. If the company is not able to recall before an inspection reveals the fault, it pays a fine per item sold, which is assumed to be much larger than the price of the product. We compute the optimal recall time that minimizes the expected cost of recall of this company. We then derive and solve a stationary limit recall problem and show that the original problem converges to it as the number of items initially sold increases to ∞. Finally, we propose two extensions of the original model and compute the optimal recall times for these. In the first extension, the expired items are inspected only if they expire earlier than expected; in the second extension, the company is able to conduct internal/private inspections on the expired items. We provide numerical examples and simulation results for all three models.

Mixed-level orthogonal arrays are basic structures in experimental design. We develop three algorithms that compute Rao- and Gilbert-Varshamov-type bounds for mixed-level orthogonal arrays. The computational complexity of the terms involved in the original combinatorial representations of these bounds can grow fast as the parameters of the arrays increase and this justifies the construction of these algorithms. The first is a recursive algorithm that computes the bounds exactly, the second is based on an asymptotic analysis, and the third is a simulation algorithm. They are all based on the representation of the combinatorial expressions that appear in the bounds as expectations involving a symmetric random walk. The Markov property of the underlying random walk gives the recursive formula to compute the expectations. A large deviation (LD) analysis of the expectations provides the asymptotic algorithm. The asymptotically optimal importance sampling (IS) of the same expectation provides the simulation algorithm. Both the LD analysis and the construction of the IS algorithm use a representation of these problems as a sequence of stochastic optimal control problems converging to a limit calculus of a variations problem. The construction of the IS algorithm uses a recently discovered method of using subsolutions to the Hamilton-Jacobi-Bellman equations associated with the limit problem.

We model an insurance system consisting of one insurance company and one reinsurance company as a stochastic process in R(2). The claim sizes {X(i)} are an iid sequence with light tails. The interarrival times {tau(i)} between claims are also iid and exponentially distributed. There is a fixed premium rate cl that the customers pay; c < c(1) of this rate goes to the reinsurance company. If a claim size is greater than R the reinsurance company pays for the claim. We study the bankruptcy of this system before it is able to handle N number of claims. It is assumed that each company has initial reserves that grow linearly in N and that the reinsurance company has a larger reserve than the insurance company. If c and c(1) are chosen appropriately, the probability of bankruptcy decays exponentially in N. We use large deviations (LD) analysis to compute the exponential decay rate and approximate the bankruptcy probability. We find that the LD analysis of the system decouples: the LD decay rate gamma of the system is the minimum of the LD decay rates of the companies when they are considered independently and separately. An analytical and numerical study of gamma as a function of (c. R) is carried out. (C) 2010 Elsevier B.V. All rights reserved.

This note describes an importance sampling (IS) algorithm to estimate buffer overflows of stable Jackson networks with a tree topology. Three new measures of service capacity and traffic in Jackson networks are introduced and the algorithm is defined in their terms. These measures are effective service rate, effective utilization and effective service-to-arrival ratio of a node. They depend on the nonempty/empty states of the queues of the network. For a node with a nonempty queue, the effective service rate equals the node’s nominal service rate. For a node i with an empty queue, it is either a weighted sum of the effective service rates of the nodes receiving traffic directly from node i , or the nominal service rate, whichever smaller. The effective utilization is the ratio of arrival rate to the effective service rate and the effective service-to-arrival ratio is its reciprocal. The rare overflow event of interest is the following: given that initially the network is empty, the system experiences a buffer overflow before returning to the empty state. Two types of buffer structures are considered: (1) a single system-wide buffer shared by all nodes, and (2) each node has its own fixed size buffer. The constructed IS algorithm is asymptotically optimal, i.e., the variance of the associated estimator decays exponentially in the buffer size at the maximum possible rate. This is proved using methods from (Dupuis et al. in Ann. Appl. Probab. 17(4):1306–1346, 2007 ), which are based on a limit Hamilton–Jacobi–Bellman equation and its boundary conditions and their smooth subsolutions. Numerical examples involving networks with as many as eight nodes are provided.

Importance sampling (IS) is a variance reduction method for simulating rare events. A recent paper by Dupuis, Wang and Sezer [Paul Dupuis, Ali Devin Sezer, Hui Wang, Dynamic importance sampling for queueing networks, Annals of Applied Probability 17 (4) (2007) 1306-1346] exploits connections between IS and stochastic games and optimal control problems to show how to design and analyze simple and efficient IS algorithms for various overflow events of tandem Jackson Networks. The present paper carries out a program parallel to the paper by Dupuis et al. for a two node tandem network whose arrival and service rates are modulated by in exogenous finite state Markov process. The overflow event we study is the following: the number of customers in the system reaches n without the system ever becoming empty, given that initially the system is empty. (C) 2008 Elsevier B.V. All rights reserved.