...
Thus, we can solve for 1/λ, which is the average interarrival time:
Example 3
Example 4
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A batch scheduling computer completes each job assigned to the CPU before starting the next job. This system can be modeled as an M/M/1 queueing system, where the average length of a job is 1/μj instructions.
Suppose there are two options available:
- One fast computer
- Ten slower computers
The single fast computer has an execution speed of C instructions/second, and each of the slower computers has an execution speed of C/10 instructions/second.
If the faster computer is used, the arrival rate of new jobs is λ. If the ten slower computers are used, each is responsible for 10% of the jobs arriving by a Poisson distribution at rate λ/10. A job is never split and handled by multiple of the 10 computers in parallel.
Determine the fraction of time each computer is busy and the average total delay for both options (1) and (2)
One Fast Computer
So:
and:
Ten Slow Computers
and:
but:
So, the configuration with ten slower computer leads to much larger delays than with the single fast computer.
In option (2), suppose any arriving or waiting job can use any available computer. What queueing model is appropriate?
This implies a ten-server queueing model, or an M/M/10 queueing system, which is more difficult to analyze.
Example 4
A total of M terminals are connected to a concentrator (i.e. a multiplexer). The output line from the concentrator operates at 1200 bits/second. Message lengths from each terminal are fixed at 28 bytes. The concentrator itself adds a 2-byte protocol header to each message, bringing the total message length to 30 bytes.
Messages from each terminal arrive to the concentrator with a Poisson distribution at a rate of 3 messages/minute.
Determine the maximum number of terminals that can be connected to the concentrator without resulting in an unstable system
Suppose X1, X2, ..., XM are independent Poisson distributed RVs with average arrival rates of λ1,λ2 , ..., λM, respectively. If this is the case, then X = X1 + X2 + ... + XM is also a Poisson distributed RV with average arrival rate λ = λ1 + λ2 + ... + λM.
Given this fact, all of the arrivals from the M terminals taken together form a Poisson arrival process with rate:
and expected service time:
Stable operation occurs from ρ < 1, which implies:
Suppose that only the original 28 bytes of each message are stored in the queue and that the 2-byte header is added during transmission. What is the average buffer space occupied if 70 terminals are attached to the connector? Include both the queue buffer and space to store a copy of the current transmission.
The system is an M/D/1 queueing system, so using the P-K formulas for M/D/1, the average messages in the system is:
and the average buffer space occupied is:
As a question for the reader, what is the answer to the above question if 99 terminals are attached?
Queueing Systems with Priority-Based Disciplines