I've been studying network of queues and I would like to model a system by a tandem state-dependent network of queues. I know Jackson network works for $M/M/$ networks, but not for queues where the arrival rate and service time are state-dependent and depends on some distribution, like a $GI/M/1$ queue. So, I would like to know if it is possible to approximate the delay and the behavior of a network whose the queues are $GI/M/1$ approximately. Each queue $Q_i$ in the system I want to model has an arrival rate $\lambda_{i1}$ with probability of $p_1$ and $\lambda_{i2}$ with probability of $p_2$, where $p_1 + p_2 = 1$. A service leaves $Q_i$ and goes to $Q_j$ with probability of $r_{ij}$. The service time of each queue $Q_i$ is exponential and depends on the length of queue. If the queue has more than $L$ jobs waiting, than the service time is $\mu_{i2}$ and $\mu_{i1}$ otherwise. I know this problem may be intractable, but I would like to know if there are some works that approximate a network of queues like that in order to turn this problem into a more tractable one.


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