Ajem Guido Janssen

Refining Square-Root Safety Staffing by Expanding Erlang C

We apply a new corrected diffusion approximation for the Erlang C formula to determine staffing levels in cost minimization and constraint satisfaction problems. These problems are motivated by large customer contact centers that are modeled as an M/M/s queue with s the number of servers or agents. The proposed staffing levels are refinements of the celebrated square-root safety-staffing rule and have the appealing property that they are as simple as the conventional square-root safety-staffing rule. In addition, we provide theoretical support for the empirical fact that square-root safety-staffing works well for moderate-sized systems.

Spatial fairness in linear random-access networks

Random-access networks may exhibit severe unfairness in throughput, in the sense that some nodes receive consistently higher throughput than others. Recent studies show that this unfairness is due to local differences in the neighborhood structure: nodes with fewer neighbors receive better access. We study the unfairness in saturated linear networks, and adapt the random-access CSMA protocol to remove the unfairness completely, by choosing the activation rates of nodes as a specific function of the number of neighbors. We then investigate the consequences of this choice of activation rates on the network-average saturated throughput, and we show that these rates perform well in non-saturated settings.

Scaled control in the QED regime

We develop many-server asymptotics in the Quality-and-Efficiency-Driven (QED) regime for models with admission control. The admission control, designed to reduce the incoming traffic in periods of congestion, scales with the size of the system. For a class of Markovian models with this scaled control, we identify the QED limits for two stationary performance measures. We also derive corrected QED approximations, generalizing earlier results for the Erlang B, C and A models. These results are useful for the dimensioning of large systems equipped with an active control policy. In particular, the corrected approximations can be leveraged to establish the optimality gaps related to square-root staffing and asymptotic dimensioning with admission control.

Balancing exposed and hidden nodes in linear wireless networks

Wireless networks equipped with the CSMA protocol are subject to collisions due to interference. For a given interference range, we investigate the tradeoff between collisions (hidden nodes) and unused capacity (exposed nodes). We show that the sensing range that maximizes throughput critically depends on the activation rate of nodes. For infinite line networks, we prove the existence of a threshold: When the activation rate is below this threshold, the optimal sensing range is small (to maximize spatial reuse). When the activation rate is above the threshold, the optimal sensing range is just large enough to preclude all collisions. Simulations suggest that this threshold policy extends to more complex linear and nonlinear topologies.

Convolution theory in a space of generalized functions

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Giant component sizes in scale-free networks with power-law degrees and cutoffs

Scale-free networks arise from power-law degree distributions. Due to the finite size of real-world networks, the power law inevitably has a cutoff at some maximum degree Δ. We investigate the relative size of the giant component S in the large-network limit. We show that S as a function of Δ increases fast when Δ is just large enough for the giant component to exist, but increases ever more slowly when Δ increases further. This gives that while the degree distribution converges to a pure power law when Δ → ∞, S approaches its limiting value at a slow pace. The convergence rate also depends on the power-law exponent τ of the degree distribution. The worst rate of convergence is found to be for the case , which concerns many of the real-world networks reported in the literature.

Wafer-based aberration metrology for lithographic systems using overlay measurements on targets imaged from phase-shift gratings

In this paper, a new methodology is presented to derive the aberration state of a lithographic projection system from wafer metrology data. For this purpose, new types of phase-shift gratings (PSGs) are introduced, with special features that give rise to a simple linear relation between the PSG image displacement and the phase aberration function of the imaging system. By using the PSGs as the top grating in a diffraction-based overlay stack, their displacement can be measured as an overlay error using a standard wafer metrology tool. In this way, the overlay error can be used as a measurand based on which the phase aberration function in the exit pupil of the lithographic system can be reconstructed. In practice, the overlay error is measured for a set of different PSG targets, after which this information serves as input to a least-squares optimization problem that, upon solving, provides estimates for the Zernike coefficients describing the aberration state of the lithographic system. In addition to a detailed method description, this paper also deals with the additional complications that arise when the method is implemented experimentally and this leads to a number of model refinements and a required calibration step. Finally, the overall performance of the method is assessed through a number of experiments in which the aberration state of the lithographic system is intentionally detuned and subsequently estimated by the new method. These experiments show a remarkably good agreement, with an error smaller than 5u2009u2009mλ, among the requested aberrations, the aberrations measured by the on-tool aberration sensor, and the results of the new wafer-based method.

A lower bound for the Erlang C formula in the Halfin---Whitt regime

One of the classical models of queueing theory is the M/M/s queue or Erlang delay model. This model has s homogeneous servers working in parallel. Customers arrive according to a Poisson process with arrival rate λ, and the service times are independent and exponentially distributed with mean 1/μ. Let the offered load be a = λ/μ and assume a < s to have a proper steady-state distribution. The most important performance characteristic for this system is the probability that a customer is delayed when arriving at the system in steady state. This probability is known as the Erlang C formula, given by (with ρ = a/s)

Note on a paper by M. Laczkovich on functions with measurable differences

This note is meant to simplify certain parts of M. Laczkovich proof of Erdos conjecture about functions with measurable differences. The (pseudo)norm occuring in Laczkovich proof is replaced by a norm (with essentially the same properties as Laczkovich norm) that admits easy manipulation. The other parts of Laczkovich proof of Erdos conjecture need hardly any alteration when Laczkovich norm is replaced by the one introduced in this note. It is further shown that the crucial property of Laczkovich norm (as given in [L], Theorem 2) can be derived from the corresponding property of our norm.