I want to examine the difference between two systems:
- Single queue with arrival rate $2\lambda$ and 2 servers with serving rate $\mu$
- A systems with 2 queues, each with arrival rate of $\lambda$ and 1 server with rate $\mu$
Intuitively, for me, it looks like these systems should be the same. But I'm trying to compare them using the performance measures and I get strange results.
What is the best way to compare these 2 systems? How should I calculate the performance measures of the 2nd system (2 queues with 1 server each)?
Thank you.
System $2$ is a bit different than what you described in the opening post. You seem to consider a join the shortest queue model with $2$ servers and identical service rates.
This system is characterized by a single Poisson arrival process with rate $2\lambda$ and two servers that each have their own queue. Servers work with rate $\mu$. A job that arrives to the system joins the queue that has the least number of jobs. If the number of jobs in both queues is equal, it joins either queue with probability $1/2$.
The analysis of this system is quite hard, below some papers that analyze the equilibrium distribution:
Haight. Two queues in parallel, Biometrika, Vol. 45, No. 3/4 (1958), pp. 401-410
Kingman. Two similar queues in parallel, Ann. Math. Statist., Vol. 32, No. 4 (1961), pp. 1314-1323
Adan, Wessels, Zijm. Analysis of the asymmetric shortest queue problem, Queueing Systems Vol. 8, No. 1 (1991), pp. 1-58.