# M/M/n Queue (Nonpreemptive with Priorities), Several Servers to One Customer, with Server Breaks

This is my first time posting here, so sorry if I missed any policy (I did try searching). I am an undergrad trying to model emergency response workloads, based on past data.

So, I have that customers arrive at a rate of $\lambda$, following a Poisson process, and are served at a rate $\mu$, following an exponential distribution, and that I have $n$ servers (i.e. M/M/n queue). I also have that each customer is assigned a priority $p = 1, 2, 3, ..., r$ with $p=1$ the highest priority. From some papers and a textbook, I have found how I would calculate waiting time, utilization, probability all servers are occupied, etc. However, they all assume that for each customer being served, only ever one server is used. They also do not take into account "break" times for servers (say, if a server takes a 12 minute break every hour). I have the amount of time each server spends on each customer, and the average break times. How would I make these considerations? Should I consider making "effective" call rates ($\lambda_{eff}$), service rates ($\mu_{eff}$), and/or number of servers ($n_{eff}$)? I have that on average the second server spends less time with the customer than the first (I have average service times per customer: first unit, second unit, etc.).

My initial thoughts would be to replace $n$ everywhere it appears in the equations for waiting time and utilization with an "effective" $n$, as in: Let $b$ = % time spent on breaks (average per server) and $\bar n$ = the average amount of servers spent on each customer. Then, $n_{eff} = (1-b)n/\bar n$. However, I cannot find much reason aside from intuition (which could be faulty) as to why this might work. It would also almost certainly result in a non-integer value for $n$, and since $n!$ appears in the regular queueing model equations (e.g. probability queue is empty), I would probably have to use the gamma function, and I don't know if that still works within the model. I cannot think how I might adjust call or service rates. I am also concerned that on average the second server might be with the customer for less time, and that is not taken into consideration with my $n_{eff}$ (should $\bar n$ be weighted by service times?). Would this in any way break the model? Is there a reference I did not find that deals with a similar problem?

Thanks.

• In the single-server setting, a common model is for the server to take an e.g. exponentially distributed vacation time after each busy period (customers may continue to arrive and wait for the server to return, or be turned away). To generalize this to a system with $c$ servers, perhaps consider starting the vacations (or "breaks") only after the system reaches $c$ customers. – Math1000 Aug 2 '16 at 23:30
• @Math1000 Thanks for the response, but I'm really having trouble understanding. Why would servers take a break once they are saturated? Isn't that exactly the opposite what you might expect, especially in emergency response? Also, I have vacation times (per hour); in my model I suppose I would input that as the expectation value of the (e.g.) exponentially distributed vacation time? Also, is there any term for when there are several servers to one customer? I cannot find any literature for it. Sorry, I am new to queueing theory and have little background in probability. – FunctionalDefect Aug 3 '16 at 23:14
• What I meant is that the first vacation takes place after a transition from $c$ customers to $c-1$ customers. There's many ways you can model this type of problem, and complex queueing models are often very difficult, if not impossible, to solve analytically. So you may want to look into simulation methods. – Math1000 Aug 3 '16 at 23:28