# Cost, disutility and loss aversion applied to a queuing system

I am currently working with a queueing system where customers enter a system and are served. Service takes a random amount of time $$\Delta t \sim F(\cdot)$$ where $$F:\mathbb{R}\to[0,1]$$ is a cumulative distribution function. Queueing theory would usually assign a fixed cost rate $$c>0$$ such that the objective function $$J:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$$ can be defines as $$J(\Delta t) = cn\Delta t$$ where $$n=0,1,2,\dots,N$$ is the total number of customers in the queue. Such a problem is minimised.

For a fixed $$n$$, we see that $$J(\Delta t)$$ is a linear function of time representing total cost. Can I refer to this as a disutility? If so then $$c$$ is the marginal disutility. Furthermore, having $$c$$ fixed does not express any value to a service that is almost completed.

To see why this is a problem, consider that during $$\Delta t$$, a service can be continued or cancelled in order to serve another customer (perhaps the customer is faster to serve or produces a more valuable outcome). Such a pre-emption of service should take into account that services are not memoryless and that older services have a better chance of being completed. As such, the disutility curve should be monotonically increasing and concave as pictured below. If no service is active then we express a linear increasing disutility as this is neutral such that no preference is expressed. This figure also shows the marginal disutilities. It is seen that for an ongoing service $$c:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$$ should be monotonically decreasing and convex.

I think these are reasonable assumptions to make in order for a decision-making agent to express appreciation of an almost completed service. This, however, raises the question of how one would describe such an agent. Compared to an agent that would assign a linear increasing disutility function to an ongoing service, one might say that our new agent is loss averse? This is because it would be more hesitant to abandon or lose a service that it views likely to be almost complete.

I am, however, hesitant to use the word loss aversion as it is a specific concept. The fact that the agent becomes hesitant to pre-empt service in fear of losing a completed service makes me think that it is appropriate in the given context.