Queueing system: M/M/2 vs 2*M/M/1

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I want to examine the difference between two systems:



  1. Single queue with arrival rate $2lambda$ and 2 servers with serving rate $mu$

  2. 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.




queueing-theory




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  • It's well-known that a single queue with multiple servers performs better.
    – Math1000
    Dec 13 '15 at 11:47










  • Can you share a link to an article or something that points that? Thanks.
    – shudima
    Dec 13 '15 at 11:50










  • I'll post a detailed answer, just wanted to mention that because it may take a while.
    – Math1000
    Dec 13 '15 at 11:51










  • By the way, see here: math.stackexchange.com/questions/100571/…
    – Math1000
    Dec 13 '15 at 11:58






  • 1




    The customer chooses the shortest. If both are equal, then he chooses randomly with 0.5 probability for each.
    – shudima
    Dec 13 '15 at 12:51














up vote
1
down vote

favorite












I want to examine the difference between two systems:



  1. Single queue with arrival rate $2lambda$ and 2 servers with serving rate $mu$

  2. 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.




queueing-theory




share|cite|improve this question






















  • It's well-known that a single queue with multiple servers performs better.
    – Math1000
    Dec 13 '15 at 11:47










  • Can you share a link to an article or something that points that? Thanks.
    – shudima
    Dec 13 '15 at 11:50










  • I'll post a detailed answer, just wanted to mention that because it may take a while.
    – Math1000
    Dec 13 '15 at 11:51










  • By the way, see here: math.stackexchange.com/questions/100571/…
    – Math1000
    Dec 13 '15 at 11:58






  • 1




    The customer chooses the shortest. If both are equal, then he chooses randomly with 0.5 probability for each.
    – shudima
    Dec 13 '15 at 12:51












up vote
1
down vote

favorite









up vote
1
down vote

favorite











I want to examine the difference between two systems:



  1. Single queue with arrival rate $2lambda$ and 2 servers with serving rate $mu$

  2. 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.




queueing-theory




share|cite|improve this question














I want to examine the difference between two systems:



  1. Single queue with arrival rate $2lambda$ and 2 servers with serving rate $mu$

  2. 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.





queueing-theory





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edited Dec 13 '15 at 11:26









AndreasT

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asked Dec 13 '15 at 11:22









shudima

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  • It's well-known that a single queue with multiple servers performs better.
    – Math1000
    Dec 13 '15 at 11:47










  • Can you share a link to an article or something that points that? Thanks.
    – shudima
    Dec 13 '15 at 11:50










  • I'll post a detailed answer, just wanted to mention that because it may take a while.
    – Math1000
    Dec 13 '15 at 11:51










  • By the way, see here: math.stackexchange.com/questions/100571/…
    – Math1000
    Dec 13 '15 at 11:58






  • 1




    The customer chooses the shortest. If both are equal, then he chooses randomly with 0.5 probability for each.
    – shudima
    Dec 13 '15 at 12:51
















  • It's well-known that a single queue with multiple servers performs better.
    – Math1000
    Dec 13 '15 at 11:47










  • Can you share a link to an article or something that points that? Thanks.
    – shudima
    Dec 13 '15 at 11:50










  • I'll post a detailed answer, just wanted to mention that because it may take a while.
    – Math1000
    Dec 13 '15 at 11:51










  • By the way, see here: math.stackexchange.com/questions/100571/…
    – Math1000
    Dec 13 '15 at 11:58






  • 1




    The customer chooses the shortest. If both are equal, then he chooses randomly with 0.5 probability for each.
    – shudima
    Dec 13 '15 at 12:51















It's well-known that a single queue with multiple servers performs better.
– Math1000
Dec 13 '15 at 11:47




It's well-known that a single queue with multiple servers performs better.
– Math1000
Dec 13 '15 at 11:47












Can you share a link to an article or something that points that? Thanks.
– shudima
Dec 13 '15 at 11:50




Can you share a link to an article or something that points that? Thanks.
– shudima
Dec 13 '15 at 11:50












I'll post a detailed answer, just wanted to mention that because it may take a while.
– Math1000
Dec 13 '15 at 11:51




I'll post a detailed answer, just wanted to mention that because it may take a while.
– Math1000
Dec 13 '15 at 11:51












By the way, see here: math.stackexchange.com/questions/100571/…
– Math1000
Dec 13 '15 at 11:58




By the way, see here: math.stackexchange.com/questions/100571/…
– Math1000
Dec 13 '15 at 11:58




1




1




The customer chooses the shortest. If both are equal, then he chooses randomly with 0.5 probability for each.
– shudima
Dec 13 '15 at 12:51




The customer chooses the shortest. If both are equal, then he chooses randomly with 0.5 probability for each.
– shudima
Dec 13 '15 at 12:51










1 Answer
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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 $2lambda$ 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.






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    up vote
    0
    down vote













    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 $2lambda$ 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.






    share|cite|improve this answer
























      up vote
      0
      down vote













      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 $2lambda$ 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.






      share|cite|improve this answer






















        up vote
        0
        down vote










        up vote
        0
        down vote









        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 $2lambda$ 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.






        share|cite|improve this answer












        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 $2lambda$ 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.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 4 '16 at 10:57









        Ritz

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