In this paper, using the discrete time model, we consider the average age of all files for a cached-files-updating system where a server generates N files and transmits them to a local cache. In order that the cached files are fresh, in each time slot the server updates files with certain probabilities. The age of one file or its age of information (AoI) is defined as the time the file stays in cache since it was last time sent to cache. Assume that each file in cache has corresponding request popularity. In this paper, we obtain the distribution function of the popularity-weighted average age over all files, which gives a complete description of this average age. For the random age of single file, both the mean and its distribution have been derived before by establishing a simple Markov chain. Using the same idea, we show that an N dimensional stochastic process can be constituted to characterize the changes of N file ages simultaneously. By solving the steady-state of the resulting process, we obtain the explicit expression of stationary probability for an arbitrary state-vector. Then, the distribution function of the popularity-weighted average age can be derived by mergering a proper set of stationary probabilities. For the possible applications, the distribution function can be utilized to calculate the probability that the average age violates certain statistical guarantee.

A multi-server parallel system dispatches the incoming job, which contains kn tasks into n servers. A job is considered to be computed if all the tasks associated with the job are processed. One job’s tasks can be encoded into at least kn “replicas” such that the job is considered to be served if any kn replicas finishing computation. In this paper, we analyze the random scheduling policy of a multi-server computing system under discrete time model in terms of Quality of Exponent (QoE), which is defined as the probability exponent that a typical job can be computed within a given number of time slots. We let kn/n be a constant. Assuming that any task of any job can be randomly dispatched by a “scheduler” to any server, and computing each task takes exactly one time slot. We divide the calculation of probability exponent into two parts, exponent of numerator and exponent of denominator. For the denominator, we give the almost exact exponent using Lagrange multiplier method, while for the numerator, an upper bound of the numerator’s exponent is provided. In addition, we also express the exponent in terms of information theoretical quantities and reconsider both of exponents in the context of large deviation theory.

The age of information (AoI) metric was proposed to measure the freshness of messages obtained at the terminal node of a status updating system. In this paper, the AoI of discrete time status updating system with probabilistic packet preemption is investigated by analyzing the steady state of a three-dimensional discrete stochastic process. Assuming the queue used in system is Ber/Geo/1/2*/η, which represents that the system size is 2 and the packet in buffer can be preempted by fresher packet with probability η. Instead of considering system’s AoI separately, we use a three-dimensional state vector (n,m,l) to simultaneously track the real time changes of the AoI, the age of packet in server, and the age of packet waiting in buffer. We give the explicit expression of system’s average AoI, and show that the average AoI of system without packet preemption is obtained by letting η=0. When η is set to 1, the mean of AoI of system having Ber/Geo/1/2* queue is obtained as well. Combining the results we have obtained and comparing them with corresponding average continuous AoIs, we propose a possible relationship between average discrete AoI with Ber/Geo/1/c queue and the average continuous AoI with M/M/1/c queue. For each of two extreme cases where η=0 and η=1, we also determine the stationary distribution of AoI using the probability generation function (PGF) method. The relations between average AoI and the packet preemption probability η, as well as AoI’s distribution curves in two extreme cases are illustrated by numerical simulations. Notice that the probabilistic packet preemption may occur, for example, in an energy harvest (EH) node of wireless sensor network, where the packet in buffer can be replaced only when the node collects enough energy. In particular, to exhibit the usefulness of our idea and methods and highlight the merits of considering discrete time systems, in this paper we give much more explanations showing that how the results about continuous AoI is derived by analyzing the corresponding discrete time system, and how the discrete age analysis is generalized to the system with multiple sources. In terms of packet service process, we also propose our idea to analyze system’s AoI when the service time distribution is relaxed to be arbitrary.