Start Over After Pre-Empt (SOAP) Protocol

Consider a job that normally requires $a$ units of time to complete. A higher authority comes and interrupts the service. The inter-arrival times of the interrupts are $X_1,X_2,\cdots$, which are the actual available service times to work on the job. Thus, if the first available service time $X_1$ is larger than $a$, the job gets completed, and the remaining service time $X_1-a$ is the first available service time for the next job. If not, the previous service is lost and service on the job starts from scratch using $X_2$ the next available service time. As before, if $X_2\ge a$, the job gets completed and the remaining service time $X_2-a$ is the first available service time for the next job. If not, the service is lost, and $X_3$ is the available service time for the job. And so on. Let $T_a$ be the time to complete the job. This is called the start over after pre-empt (SOAP) protocol. An example of such interrupts is power outages. After a power outage interrupt the partial service on a job running on a computer is lost and service on the job has to start again. We study properties of $T_a$ and $T_{a,b}$, the time to complete two jobs normally requiring times $a,b$, and give some guidance whether $E\left(T_{a,b}\right)\le E\left(T_{b,a}\right)$ when $a<b$. We obtain the asymptotic distribution of a stochastic process based on $T_a$ which has interesting independent increments of a new type. My frustrations standing in a queue in India, where a supervisor interrupts and the service on my job has to start again, were my inspiration for the protocol named here as SOAP. A careful referee pointed out that the SOAP protocol is not new in the queues literature. It goes by the name RESTART and has variants called RESUME and REPLACE.