Split Poisson arrival according to output of Optimization problem

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I have 3 nodes with Poisson arrival rates $lambda_1, lambda_2, lambda_3$. As shown. The arrivals (packets) are sent to either of the two nodes according to the binary optimization varible $x_ij$. For example, if $x_11=1$ items arriving at rate $lambda_1$ is sent to the first queue (top in figure).



Now, the optimization problem i.e. where to send the packets depends on the transmission delay experienced by the packets. So, I am thinking that at the destination queues, the arrivals are no longer independent and the arrivals can no longer be modeled as a Poisson distribution. Since the optimization variables are dependent on each other, this cannot be treated as a probabilistic Poisson splitting. Is this correct? Should I model it as a general distribution?







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    enter image description here



    I have 3 nodes with Poisson arrival rates $lambda_1, lambda_2, lambda_3$. As shown. The arrivals (packets) are sent to either of the two nodes according to the binary optimization varible $x_ij$. For example, if $x_11=1$ items arriving at rate $lambda_1$ is sent to the first queue (top in figure).



    Now, the optimization problem i.e. where to send the packets depends on the transmission delay experienced by the packets. So, I am thinking that at the destination queues, the arrivals are no longer independent and the arrivals can no longer be modeled as a Poisson distribution. Since the optimization variables are dependent on each other, this cannot be treated as a probabilistic Poisson splitting. Is this correct? Should I model it as a general distribution?







    share|cite|improve this question






















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      down vote

      favorite









      up vote
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      down vote

      favorite











      enter image description here



      I have 3 nodes with Poisson arrival rates $lambda_1, lambda_2, lambda_3$. As shown. The arrivals (packets) are sent to either of the two nodes according to the binary optimization varible $x_ij$. For example, if $x_11=1$ items arriving at rate $lambda_1$ is sent to the first queue (top in figure).



      Now, the optimization problem i.e. where to send the packets depends on the transmission delay experienced by the packets. So, I am thinking that at the destination queues, the arrivals are no longer independent and the arrivals can no longer be modeled as a Poisson distribution. Since the optimization variables are dependent on each other, this cannot be treated as a probabilistic Poisson splitting. Is this correct? Should I model it as a general distribution?







      share|cite|improve this question












      enter image description here



      I have 3 nodes with Poisson arrival rates $lambda_1, lambda_2, lambda_3$. As shown. The arrivals (packets) are sent to either of the two nodes according to the binary optimization varible $x_ij$. For example, if $x_11=1$ items arriving at rate $lambda_1$ is sent to the first queue (top in figure).



      Now, the optimization problem i.e. where to send the packets depends on the transmission delay experienced by the packets. So, I am thinking that at the destination queues, the arrivals are no longer independent and the arrivals can no longer be modeled as a Poisson distribution. Since the optimization variables are dependent on each other, this cannot be treated as a probabilistic Poisson splitting. Is this correct? Should I model it as a general distribution?









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      share|cite|improve this question




      share|cite|improve this question










      asked Aug 25 at 7:57









      Niloy Saha

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