How to analyze this type of queue
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The setup is as follows:
Families arrive at a taxi stand according to a Poisson process with rate $lambda$. An arriving family finding $N$ other families waiting for a taxi does not wait. Taxis arrive at the taxi stand according to a Poisson process with rate $mu$. A taxi finding $M$ other taxis waiting does not wait. Derive expressions for the proportion of time are there no families waiting, and the proportion of time are there no taxis waiting.
From what I have learned so far, this appears to me to be a $M/M/c/K$ queue, where there are $c$ servers (taxis) and system capacity $K$ (since there are at most $N$ families in line). Then when finding the proportion of time are there no taxis waiting, the taxis would become the "customers" and the families the "servers". However, the $M/M/c/K$ queue has the condition that $cleq K$, which is not specified in this question. I am wondering if I may proceed as if it were an $M/M/c/K$ queue in both scenarios, then find the balance equations to derive $pi_0$ in both case.
probability markov-process queueing-theory
asked Nov 20 at 7:54
jofl
39439
add a comment |
up vote
1
down vote
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The setup is as follows:
Families arrive at a taxi stand according to a Poisson process with rate $lambda$. An arriving family finding $N$ other families waiting for a taxi does not wait. Taxis arrive at the taxi stand according to a Poisson process with rate $mu$. A taxi finding $M$ other taxis waiting does not wait. Derive expressions for the proportion of time are there no families waiting, and the proportion of time are there no taxis waiting.
From what I have learned so far, this appears to me to be a $M/M/c/K$ queue, where there are $c$ servers (taxis) and system capacity $K$ (since there are at most $N$ families in line). Then when finding the proportion of time are there no taxis waiting, the taxis would become the "customers" and the families the "servers". However, the $M/M/c/K$ queue has the condition that $cleq K$, which is not specified in this question. I am wondering if I may proceed as if it were an $M/M/c/K$ queue in both scenarios, then find the balance equations to derive $pi_0$ in both case.
probability markov-process queueing-theory
asked Nov 20 at 7:54
jofl
39439
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
The setup is as follows:
Families arrive at a taxi stand according to a Poisson process with rate $lambda$. An arriving family finding $N$ other families waiting for a taxi does not wait. Taxis arrive at the taxi stand according to a Poisson process with rate $mu$. A taxi finding $M$ other taxis waiting does not wait. Derive expressions for the proportion of time are there no families waiting, and the proportion of time are there no taxis waiting.
From what I have learned so far, this appears to me to be a $M/M/c/K$ queue, where there are $c$ servers (taxis) and system capacity $K$ (since there are at most $N$ families in line). Then when finding the proportion of time are there no taxis waiting, the taxis would become the "customers" and the families the "servers". However, the $M/M/c/K$ queue has the condition that $cleq K$, which is not specified in this question. I am wondering if I may proceed as if it were an $M/M/c/K$ queue in both scenarios, then find the balance equations to derive $pi_0$ in both case.
probability markov-process queueing-theory
asked Nov 20 at 7:54
jofl
39439
The setup is as follows:
Families arrive at a taxi stand according to a Poisson process with rate $lambda$. An arriving family finding $N$ other families waiting for a taxi does not wait. Taxis arrive at the taxi stand according to a Poisson process with rate $mu$. A taxi finding $M$ other taxis waiting does not wait. Derive expressions for the proportion of time are there no families waiting, and the proportion of time are there no taxis waiting.
From what I have learned so far, this appears to me to be a $M/M/c/K$ queue, where there are $c$ servers (taxis) and system capacity $K$ (since there are at most $N$ families in line). Then when finding the proportion of time are there no taxis waiting, the taxis would become the "customers" and the families the "servers". However, the $M/M/c/K$ queue has the condition that $cleq K$, which is not specified in this question. I am wondering if I may proceed as if it were an $M/M/c/K$ queue in both scenarios, then find the balance equations to derive $pi_0$ in both case.
probability markov-process queueing-theory
probability markov-process queueing-theory
asked Nov 20 at 7:54
jofl
39439
asked Nov 20 at 7:54
jofl
39439
asked Nov 20 at 7:54
jofl
39439
asked Nov 20 at 7:54
jofl
39439
asked Nov 20 at 7:54
jofl
39439
39439
add a comment |
add a comment |
1 Answer
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1
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Hint:
I think you could rewrite the question to an $M/M/1$ queue with capacity $N+M$ and baulking, saying something like:
Customers arrive at a queue according to a Poisson process with rate $lambda$. A customer finding $N+M$ other customers waiting does not wait. Service times are exponentially distributed with rate parameter $mu$. Derive expressions for the proportion of time are there are $M$ or fewer customers waiting, and the proportion of time there are $M$ or more customers waiting.
answered Nov 20 at 8:28
Henry
97.3k474155
Thanks. I have derived $pi_j$ in this $M/M/1/N+M$ scenario, then found $sum_{j=0}^M pi_j$ to answer the first part, and $sum_{j=M}^{N+M} pi_j$ for the second. I am wondering though how to define a "customer" in this setting though: would it still be the families arriving or would it be the pairing of a taxi with a family?
– jofl
Nov 21 at 1:09
1
Here I think having $k$ customers in the queue would be the equivalent of having $k-M$ families waiting if $kge M$, or of $M-k$ taxis waiting if $k le M$
– Henry
Nov 21 at 8:18
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Hint:
I think you could rewrite the question to an $M/M/1$ queue with capacity $N+M$ and baulking, saying something like:
Customers arrive at a queue according to a Poisson process with rate $lambda$. A customer finding $N+M$ other customers waiting does not wait. Service times are exponentially distributed with rate parameter $mu$. Derive expressions for the proportion of time are there are $M$ or fewer customers waiting, and the proportion of time there are $M$ or more customers waiting.
answered Nov 20 at 8:28
Henry
97.3k474155
Thanks. I have derived $pi_j$ in this $M/M/1/N+M$ scenario, then found $sum_{j=0}^M pi_j$ to answer the first part, and $sum_{j=M}^{N+M} pi_j$ for the second. I am wondering though how to define a "customer" in this setting though: would it still be the families arriving or would it be the pairing of a taxi with a family?
– jofl
Nov 21 at 1:09
1
Here I think having $k$ customers in the queue would be the equivalent of having $k-M$ families waiting if $kge M$, or of $M-k$ taxis waiting if $k le M$
– Henry
Nov 21 at 8:18
add a comment |
up vote
1
down vote
Hint:
I think you could rewrite the question to an $M/M/1$ queue with capacity $N+M$ and baulking, saying something like:
Customers arrive at a queue according to a Poisson process with rate $lambda$. A customer finding $N+M$ other customers waiting does not wait. Service times are exponentially distributed with rate parameter $mu$. Derive expressions for the proportion of time are there are $M$ or fewer customers waiting, and the proportion of time there are $M$ or more customers waiting.
answered Nov 20 at 8:28
Henry
97.3k474155
Thanks. I have derived $pi_j$ in this $M/M/1/N+M$ scenario, then found $sum_{j=0}^M pi_j$ to answer the first part, and $sum_{j=M}^{N+M} pi_j$ for the second. I am wondering though how to define a "customer" in this setting though: would it still be the families arriving or would it be the pairing of a taxi with a family?
– jofl
Nov 21 at 1:09
1
Here I think having $k$ customers in the queue would be the equivalent of having $k-M$ families waiting if $kge M$, or of $M-k$ taxis waiting if $k le M$
– Henry
Nov 21 at 8:18
add a comment |
up vote
1
down vote
up vote
1
down vote
Hint:
I think you could rewrite the question to an $M/M/1$ queue with capacity $N+M$ and baulking, saying something like:
Customers arrive at a queue according to a Poisson process with rate $lambda$. A customer finding $N+M$ other customers waiting does not wait. Service times are exponentially distributed with rate parameter $mu$. Derive expressions for the proportion of time are there are $M$ or fewer customers waiting, and the proportion of time there are $M$ or more customers waiting.
answered Nov 20 at 8:28
Henry
97.3k474155
Hint:
I think you could rewrite the question to an $M/M/1$ queue with capacity $N+M$ and baulking, saying something like:
Customers arrive at a queue according to a Poisson process with rate $lambda$. A customer finding $N+M$ other customers waiting does not wait. Service times are exponentially distributed with rate parameter $mu$. Derive expressions for the proportion of time are there are $M$ or fewer customers waiting, and the proportion of time there are $M$ or more customers waiting.
answered Nov 20 at 8:28
Henry
97.3k474155
answered Nov 20 at 8:28
Henry
97.3k474155
answered Nov 20 at 8:28
Henry
97.3k474155
answered Nov 20 at 8:28
Henry
97.3k474155
97.3k474155
Thanks. I have derived $pi_j$ in this $M/M/1/N+M$ scenario, then found $sum_{j=0}^M pi_j$ to answer the first part, and $sum_{j=M}^{N+M} pi_j$ for the second. I am wondering though how to define a "customer" in this setting though: would it still be the families arriving or would it be the pairing of a taxi with a family?
– jofl
Nov 21 at 1:09
1
Here I think having $k$ customers in the queue would be the equivalent of having $k-M$ families waiting if $kge M$, or of $M-k$ taxis waiting if $k le M$
– Henry
Nov 21 at 8:18
add a comment |
Thanks. I have derived $pi_j$ in this $M/M/1/N+M$ scenario, then found $sum_{j=0}^M pi_j$ to answer the first part, and $sum_{j=M}^{N+M} pi_j$ for the second. I am wondering though how to define a "customer" in this setting though: would it still be the families arriving or would it be the pairing of a taxi with a family?
– jofl
Nov 21 at 1:09
1
Here I think having $k$ customers in the queue would be the equivalent of having $k-M$ families waiting if $kge M$, or of $M-k$ taxis waiting if $k le M$
– Henry
Nov 21 at 8:18
Thanks. I have derived $pi_j$ in this $M/M/1/N+M$ scenario, then found $sum_{j=0}^M pi_j$ to answer the first part, and $sum_{j=M}^{N+M} pi_j$ for the second. I am wondering though how to define a "customer" in this setting though: would it still be the families arriving or would it be the pairing of a taxi with a family?
– jofl
Nov 21 at 1:09
Thanks. I have derived $pi_j$ in this $M/M/1/N+M$ scenario, then found $sum_{j=0}^M pi_j$ to answer the first part, and $sum_{j=M}^{N+M} pi_j$ for the second. I am wondering though how to define a "customer" in this setting though: would it still be the families arriving or would it be the pairing of a taxi with a family?
– jofl
Nov 21 at 1:09
1
1
Here I think having $k$ customers in the queue would be the equivalent of having $k-M$ families waiting if $kge M$, or of $M-k$ taxis waiting if $k le M$
– Henry
Nov 21 at 8:18
Here I think having $k$ customers in the queue would be the equivalent of having $k-M$ families waiting if $kge M$, or of $M-k$ taxis waiting if $k le M$
– Henry
Nov 21 at 8:18
add a comment |
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