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Box 31.9 G/M/1 Queue

1.  Parameters:
E[τ] = mean interarrival time
µ = service rate in jobs per unit time
If is the Laplace transform of the probability density function of the interarrival time τ, then let be the solution of the equation
2.  Traffic intensity: ρ = 1/(E[τ] µ)
3.  The system is stable if the traffic intensity ρ is less than 1.
4.  Probability of zero jobs in the system: ρ0 = 1 – ρ
5.  Probability of n jobs in the system:
6.  Mean number of jobs in the system:
7.  Variance of number of jobs in the system:
8.  Cumulative distribution function of the response time:
9.  Mean response time:
10.  Variance of response time:
11.  Probability distribution function of the waiting time:
12.  Mean waiting time:
13.  Variance of waiting time:
14.  q-Percentile of the response time: E[r]ln[100/(100 – q)]
15.  90-Percentile of the response time: E[r]ln[101 = 2.3E[r]
16.  q-Percentile of the waiting time: max(0,(E[w]/)ln[100/(100 – q)]). At low traffic intensities, the second term in this expression can be negative. The correct q-percentile for those cases is 0.
17.  90-Percentile of the waiting time: max(0,(E[w]/)ln(10)). At low traffic intensities, the second term in this expression can be negative. The correct 90-percentile for those cases is 0.
18.  Probability of finding n or more jobs in the system:

For Poisson arrivals, and all formulas become identical to those for M/M/1 queues.

Box 31.10 G/G/m Queue

1.  Parameters:
E[τ] = mean interarrival time
λ = arrival rate, = 1/E[τ]
E[s] = mean service time per job
µ = service rate, = 1/E[s]
2.  Traffic intensity: ρ = λ/(mµ)
3.  The system is stable if traffic intensity ρ is less than 1.
4.  Mean number of jobs in service: E[ns] = mρ
5.  Mean number of jobs in the system: E[n] = E[nq] + mρ
6.  Variancen of number of jobs in the system: Var[n] = Var[nq] + Var[ns]
7.  Mean response time: E[r] = E[w] + E[s]. Alternately, E[r] = E[n]/λ
8.  Variance of the response time: Var[r] = Var[w] + Var[s]
9.  Mean waiting time: E[w] = E[nq]/λ

EXERCISES

31.1  Consider a single-server system with discouraged arrivals in which the arrival rate is only λ/(n + 1) when there are n jobs in the system. The interarrival times, as well as the service time, are independent and identically distributed with an exponential distribution. Using a birth-death process model for this system, develop expressions for the following:
a.  State probability pnof n jobs in the system
b.  State probability ρ0 of the system being idle c. Mean number of jobs in the system, E[n]
d.  Effective arrival rate λ’
e.  Mean response time E[r]
31.2  Consider an M/M/∞ system with an infinite number of servers. In such a system, all arriving jobs begin receiving service immediately. The service rate is when there are n jobs in the system. Using a birth-death process model for this system, draw a state transition diagram for this system and develop expressions for the following:
a.  State probability pn of n jobs in the system
b.  State probability p0 of the system being idle
c.  Mean number of jobs in the system, E[n]
d.  Variance of number of jobs in the system, Var[n]
e.  Mean response time E[r]
31.3  The average response time on a database system is 3 seconds. During a 1-minute observation interval, the idle time on the system was measured to be 10 seconds. Using an M/M/1 model for the system, determine the following:
a.  System utilization
b.  Average service time per query
c.  Number of queries completed during the observation interval3
d.  Average number of jobs in the system
e.  Probability of number of jobs in the system being greater than 10
f.  90-percentile response time
g.  90-percentile waiting time
31.4  A storage system consists of three disk drives sharing a common queue. The average time to service an I/O request is 50 milliseconds. The I/O requests arrive to the storage system at the rate of 30 requests per second. Using an M/M/3 model for this system, determine the following:
a.  Average disk drive utilization
b.  Probability of the system being idle, p0
c.  Probability of queueing,
d.  Average number of jobs in the system, E[n]
e.  Average number of jobs waiting in the queue, E[nq]
f.  Mean response time E[r]
g.  Variance of the response time
h.  90-percentile of the waiting time
31.5  Complete Exercise 31.4 again assuming that a separate queue is maintained for each disk drive of the system. Also, assume the same total arrival rate.
31.6  Consider the problem of Example 31.3. The university is also considering the option to replace its computer system with a newer version that is twice as fast. The students will be able to finish their jobs within 10 minutes on an average. Would this system satisfy students’ demands?
31.7  In the computer center problem of Example 31.3, space limitations do not allow more terminals to be installed at the current facility. If another identical computer facility is built in another part of the campus, will it help satisfy the students’ requirements?
31.8  Assuming that there are only four buffers in the system of Exercise 31.4, determine the following:
a.  Probability pn of n jobs in the system for n = 0,1,...,4
b.  Mean number of jobs in the system, E[n]
c.  Mean number of requests in the queue, E[nq]
d.  Variance of the number of jobs in the system, Var[n]
e.  Effective arrival rate λ’
f.  Request loss rate
g.  Drive utilization
h.  Mean response time E[r]
31.9  An M/M/m//K queue with a finite population of size K can be modeled as a birth-death process using the following arrival and service rates:


Derive an expression for the probability pn of n jobs in the system, average throughput, and average response time of the system.
31.10  An M/M/m/B/K queue with B buffers and finite population of size K can be modeled as a birth-death process using the following arrival and service rates:


Derive an expression for the probability pn of n jobs in the system, average throughput, and average response time of the system.


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