Collaborating queues: large service network and a limit order book

E-thesis pagination differs from hardbound copy kept in the Manuscripts Department, Cambridge University Library.We analyse the steady-state behaviour of two different models with collaborating queues: that is, models in which "customers" can be served by many types of "servers", and "servers" can process many types of "customers". The first example is a large-scale service system, such as a call centre. Collaboration is the result of cross-trained staff attending to several different types of incoming calls. We first examine a load-balancing policy, which aims to keep servers in different pools equally busy. Although the policy behaves order-optimally over fixed time horizons, we show that the steady-state distribution may fail to be tight on the diffusion scale. That is, in a family of ever-larger networks whose arrival rates grow as O(r) (where r is a scaling parameter growing to infinity), the sequence of steady-state deviations from equilibrium scaled down by sqrt(r) is not tight. We then propose a different policy, for which we show that the sequence of invariant distributions is tight on the r^(1/2+epsilon) scale, for any epsilon > 0. For this policy we conjecture that tightness holds on the diffusion scale as well. The second example models a limit order book, a pricing mechanism for a single-commodity market in which buyers (respectively sellers) are prepared to wait for the price to drop (respectively rise). We analyse the behaviour of a simplified model, in which the arrival events are independent of each other and the state of the limit order book. The system can be represented by a queueing model, with "customers" and "servers" corresponding to bids and asks; the roles of customers and servers are symmetric. We show that, with probability 1, the price interval breaks up into three regions. At small (respectively large) prices, only finitely many bid (respectively ask) orders ever get fulfilled, while in the middle region all orders eventually clear. We derive equations which define the boundaries between these regions, and solve them explicitly in the case of iid uniform arrivals to obtain numeric values of the thresholds. We derive a heuristic for the distribution of the highest bid (respectively lowest ask), and present simulation data confirming it.This work was supported by the US National Science Foundation Graduate Research Fellowship

High-dimensional estimation with geometric constraints

Consider measuring an n-dimensional vector x through the inner product with several measurement vectors, a_1, a_2, ..., a_m. It is common in both signal processing and statistics to assume the linear response model y_i = + e_i, where e_i is a noise term. However, in practice the precise relationship between the signal x and the observations y_i may not follow the linear model, and in some cases it may not even be known. To address this challenge, in this paper we propose a general model where it is only assumed that each observation y_i may depend on a_i only through . We do not assume that the dependence is known. This is a form of the semiparametric single index model, and it includes the linear model as well as many forms of the generalized linear model as special cases. We further assume that the signal x has some structure, and we formulate this as a general assumption that x belongs to some known (but arbitrary) feasible set K. We carefully detail the benefit of using the signal structure to improve estimation. The theory is based on the mean width of K, a geometric parameter which can be used to understand its effective dimension in estimation problems. We determine a simple, efficient two-step procedure for estimating the signal based on this model -- a linear estimation followed by metric projection onto K. We give general conditions under which the estimator is minimax optimal up to a constant. This leads to the intriguing conclusion that in the high noise regime, an unknown non-linearity in the observations does not significantly reduce one's ability to determine the signal, even when the non-linearity may be non-invertible. Our results may be specialized to understand the effect of non-linearities in compressed sensing.Comment: This version incorporates minor revisions suggested by referee

Matrix-Ball Construction of affine Robinson-Schensted correspondence

International audienceIn his study of Kazhdan-Lusztig cells in affine type A, Shi has introduced an affine analog of Robinson- Schensted correspondence. We generalize the Matrix-Ball Construction of Viennot and Fulton to give a more combi- natorial realization of Shi's algorithm. As a biproduct, we also give a way to realize the affine correspondence via the usual Robinson-Schensted bumping algorithm. Next, inspired by Honeywill, we extend the algorithm to a bijection between extended affine symmetric group and triples (P, Q, ρ) where P and Q are tabloids and ρ is a dominant weight. The weights ρ get a natural interpretation in terms of the Affine Matrix-Ball Construction. Finally, we prove that fibers of the inverse map possess a Weyl group symmetry, explaining the dominance condition on weights