# Throughput of Buffer with Dependent Service Times
###### Abstract  
We study the throughput and losses of a buffer with stochastically dependent service times. Such dependence occurs not only in packet buffers within TCP/IP networks but also in many other queuing systems. We conduct a comprehensive, time-dependent analysis, which includes deriving formulae for the count of packets processed and lost over an arbitrary period, the temporary intensity of output traffic, the temporary intensity of packet losses, buffer throughput, and loss probability. The model considered enables mimicking any packet interarrival time distribution, service time distribution, and correlation between service times. The analytical findings are accompanied by numerical computations that demonstrate the influence of various factors on buffer throughput and losses. These results are also verified through simulations. p 2025 8 34  
on the stationary analysis of a few basic characteristics, such as queue size and sojourn time (more details will be provided in the next section). In this paper, we conduct a comprehensive analysis of the throughput-related characteristics of a buffer model with dependent service times and finite buffer capacity. Specifically, we derive formulae for the count of packets processed and lost over an arbitrary interval, the time-dependent output traffic intensity, throughput and loss probability, and the time-dependent loss intensity. Naturally, all throughput-related characteristics are especially important from a networking perspective, where they constitute crucial performance metrics in addition to queueing-induced delay. The queuing model considered consists of a renewal input process with general distribution of the time intervals between consecutive packets. The consecutive packet transmission times are modeled using a Markovian service process. This service model combines analytical accessibility with the ability to model any real service time distribution with arbitrary precision and to mimic any correlation function of service times. Additionally, a finite buffer is considered, which is essential given real networking buffers and packet losses resulting from buffer overflows. In addition to theorems and formulae, a set of numerical examples is presented. Rather than using fixed arrival and service processes, we consider a class of arrival processes and a class of service processes in these examples, with continuously increasing standard deviation of the interval between arrivals and continuously increasing correlation coefficients of service times. We demonstrate how these features influence throughput-related characteristics in both time-dependent and stationary cases. Finally, the analytical results are validated using discrete-event simulations performed in the OMNeT++ environment [3]. In detail, the key contributions of this article consist of the following:  
1. A theorem on the expected count of packets processed in \((0,t)\) (Theorem 1);
2. A formula on the output traffic intensity at \(t\) (Corollary 1);
3. A formula for the buffer throughput (Corollary 2);
4. A theorem on the expected count of lost packets in \((0,t)\) (Theorem 2);
5. A formula on the loss intensity at \(t\) (Corollary 3);
6. A formula for the overall loss probability (Corollary 4). Finally, it should be underlined that the goal of this paper is not to study a precise model of TCP/IP traffic and its interaction with the buffer. Such a model would have to incorporate other features of traffic that are not present in this paper. The main purpose of this paper is to study how the dependence of service times influences the buffer throughput. Therefore, for instance, the correlation of interarrival times--which is also present in TCP/IP traffic and certainly influences the throughput--is intentionally omitted. Including multiple factors, each of which causes the degradation of the throughput, would make it difficult to separate the impact of each factor.