Molecular sequences obtained at different sampling times from populations of rapidly evolving pathogens and from ancient subfossil and fossil sources are increasingly available with modern sequencing technology. Here, we present a Bayesian statistical inference approach to the joint estimation of mutation rate and population size that incorporates the uncertainty in the genealogy of such temporally spaced sequences by using Markov chain Monte Carlo (MCMC) integration. The Kingman coalescent model is used to describe the time structure of the ancestral tree. We recover information about the unknown true ancestral coalescent tree, population size, and the overall mutation rate from temporally spaced data, that is, from nucleotide sequences gathered at different times, from different individuals, in an evolving haploid population. We briefly discuss the methodological implications and show what can be inferred, in various practically relevant states of prior knowledge. We develop extensions for exponentially growing population size and joint estimation of substitution model parameters. We illustrate some of the important features of this approach on a genealogy of HIV-1 envelope (env) partial sequences.

Consider a network where two routes are available for users wishing to travel from a source to a destination. On one route (which could be viewed as private transport) service slows as traffic increases. On the other (which could be viewed as public transport) the service frequency increases with demand. The Downs-Thomson paradox occurs when improvements in service produce an overall decline in performance as user equilibria adjust. Using the model proposed by Calvert [10], with a ·|M|1 queue corresponding to the private transport route, and a bulk-service infinite server queue modelling the public transport route, we give a complete analysis of this system in the setting of probabilistic routing. We obtain the user equilibria (which are not always unique), and determine their stability.

We consider initially two parallel routes, each of two queues in tandem, with arriving customers choosing the route giving them the shortest expected time in the system, given the queue lengths at the customer's time of arrival. All interarrival and service times are exponential. We then augment this network to obtain a Wheatstone bridge, in which customers may cross from one route to the other between queues, again choosing the route giving the shortest expected time in the system, given the queue lengths ahead of them. We find that Braess's paradox can occur: namely in equilibrium the expected transit time in the augmented network, for some service rates, can be greater than in the initial network.

Consider a system of two queues in parallel, one of which is a ·|M|1 single-server infinite capacity queue, and the other a ·|G (N ) |∞ batch service queue. A stream of general arrivals choose which queue to join, after observing the current state of the system, and so as to minimize their own expected delay. We show that a unique user equilibrium (user optimal policy) exists and that it possesses various monotonicity properties, using sample path and coupling arguments. This is a very simplified model of a transportation network with a choice of private and public modes of transport. Under probabilistic routing (which is equivalent to the assumption that users have knowledge only of the mean delays on routes), the network may exhibit the Downs-Thomson paradox observed in transportation networks with expected delay increasing as the capacity of the ·|M|1 queue (private transport) is increased. We give examples where state-dependent routing mitigates the DownsThomson effect observed under probabilistic routing, and providing additional information on the state of the system to users reduces delay considerably.Mathematics Subject Classification (2010) 90B15 · 60K25 · 90B20 · 91A25 · 91A10 · 91A13

We review and develop Bayesian statistical methods for recovering genealogical structure, population size and mutation rates from radiocarbon-dated fossil mtDNA sequence data. It is possible to obtain ages for fossil DNA sequences and their common ancestors, by fitting a population-genetic model. We describe the observation model and show how uncertainty in reconstructed parameter values may be quantified via sample-based inference. We give an example, in which errors arising from radiocarbon calibration of fossil sequences are dominated by uncertainty in the genealogy and associated population parameters. We do not discuss likely model mispecification in any detail.