Model and Assumptions. The entities of the KA group are represented as the vertices V in a (not necessarily connected) graph G:{V,E}. An edge is attributed between two vertices, representing a communication link, if and only if the associated two entities are within radio range of each other. In this scenario, as could also be the case in a secret army squad, each entity knows the IDs of the rest of the participants, but this constraint may also be relaxed. We assume that all nodes participating to the KA process are honest, but consider node failures, due to crashes or mobility, and model them using node and edge failures in G. An adversary is allowed to eavesdrop on the entire communication along the graph; Byzantine adversaries will be addressed in future work. Thus, routing within paths on G (only among members of the group) is considered reliable (when connectivity is superior to the number of crashes).
Model and Assumptions. We consider a call center with two classes of customer, namely contract (type 1) and non- contract (type 2) customers. Contract customers have a service level agreement that the firm must adhere to. We assume that, once the contract is in place, the firm must determine a policy that satisfies the agreement, while trying to serve non-contract customers as well. Customers within each class are homogeneous and are assumed to arrive according to a general renewal process with independent identically distributed interarrival times, with mean 1/λi, for i = 1, 2. The arrival rates for both classes are assumed to be constant; thus, we do not examine the problem of gaining and/or losing customers. Also, we do not consider abandonment from the queue. There are N servers that work in parallel. Both types of customer are assumed to have the same service time distribution. Further, service times have the same distribution across all servers. All interarrival times and service times are assumed to be mutually independent and independent of the state of the system. The delay percentile contract that we consider is specified by two parameters, D and γ. The contract requires that γ × 100% of all contract (type 1) customers wait no more than D time units. A policy defines at each time point which server is working on which queue. A feasible policy must satisfy the contract when the time-average over all departed customers is considered. We restrict attention to feasible policies that are First-Come-First- Serve (FCFS) within customer class. Such policies are often referred to as “Head-of-the-Line” service. Finally, we only consider policies that are stable. Rather than analyzing the actual system in terms of customer delay or queue length, we will concentrate on analysis of the system under heavy-traffic. While analysis of policies for the system in “regular” traffic is difficult, considerable simplifications occur under heavy- traffic. We can thus obtain tractable results, which will then provide insights that will be applied to the original regular traffic system. We define an impulse-exponential random variable as an exponential random variable with a possible pulse at zero. Instead of explicitly considering a particular heavy-traffic limit, we prove our results under any heavy-traffic limit that results in an impulse-exponential steady-state limit for scaled queue length (see Xxxxxx and Xxxxx 2004 for the formalities). Such scalings include the “conventional” heavy-traf...
Model and Assumptions. We consider a call center with two classes of customer, namely contract (type 1) and non-contract (type 2) customers. Customers within each class are homogeneous and are assumed to arrive to queue i according to a general renewal process with independent identically distributed interarrival times with mean 1/λi and squared coefficient of variation c2 , i = 1, 2. Let Λ = λ1 + λ2 be the total arrival rate and c2 = ai a a1 a2 (λ1c2 + λ2c2 )/Λ be the averaged squared coefficient of variation over both arrival processes. Also let pi = λi/Λ be the proportion of arrivals to class i, i = 1, 2. The arrival rates for both classes are assumed to be constant; thus, we do not examine the problem of gaining and/or losing customers. Also, we do not consider abandonment from the queue. Both extensions are discussed briefly in Section 4. s There are N servers that work in parallel. Both types of customer are assumed to have the same service time distribution (we discuss relaxation of this assumption later). Further, service times have the same distribution across all servers with mean 1/µ and squared coefficient of variation c2. Let ρi = λi/(Nµ) be the traffic intensity of class i customers and let ρ = ρ1 + ρ2 be the system traffic intensity; we assume that ρ < 1. All interarrival times and service times are assumed to be mutually independent and independent of the state of the system. The delay percentile contract that we consider is specified by two parameters, D and γ. The contract requires that γ × 100% of all contract (type 1) customers wait no more than D time units. A policy defines at each time point which server is working on which queue. A feasible policy must satisfy the contract when the time-average over all departed customers is considered. We restrict attention to feasible policies that are First-Come-First-Serve (FCFS) within customer class. Such policies are often referred to as “Head-of-the- Line” service; we return to discussion of this assumption in Section 4. Finally, we only consider policies that are stable, in that queue length and delay (see below) have proper limiting distributions in a time-average sense (see, e.g., p. 237, Xxxxx 1989) yielding steady-state random variables. Returning to notational definitions, let Xi(t) be the number of customers of type i in the system at time t, i = 1, 2, t ≥ 0. Further, let X(t) = X1(t) + X2(t) be the total number of customers in the system. Note that X(t) is completely determined by the arrival and service i...
