A Note on the M/G/1 Queue with Server Vacations

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A Note on the M/G/1 Queue with Server Vacations

• S.W. Fuhrmann
• Operations Research, Vol. 32, No. 6 (Nov. Dec.,
  1984), 1368-1373

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Outline

• Problem Definition
• Notation
• The M/G/1 Queue with Server Vacations
• Extensions

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Outline

• Problem Definition
• Notation
• The M/G/1 Queue with Server Vacations
• Extensions

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Problem Definition

• Server begins a vacation of random length each
  time the system becomes empty.
• If he returns to find no customers waiting,
  begins another vacation immediately.
• If he returns to find customers waiting, works
  until the system empties and then begins another
  vacation.

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Problem Definition (contd)

• The number of customers present in the system at
  a random point in time in equilibrium
• The number of Poisson arrivals during a time
  interval that is distributed as the equilibrium
  forward recurrence time (residual life) of a
  vacation.
• The number of customers present at a random point
  in time in equilibrium in the corresponding
  standard M/G/1 queuing system.

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Outline

• Problem Definition
• Notation
• The M/G/1 Queue with Server Vacations
• Extensions

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Notation

• ? the arrival rate of customers to the system
• V(.) the distribution function of the length of
  a vacation
• v the mean value of V(.)
• R(.) the distribution function of the
  equilibrium forward recurrence time of a vacation
  (Cox and Lewis,1968. The Statistical analysis of
  Series of Events. Section 4.3)

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Notation (contd)

• pi(.)the p.g.f for the number of customers that
  a random departing customer leaves behind in
  system i.
• Wi(.) the distribution function of the sojourn
  time that a random customer experiences in system
  i, under a FIFO queuing discipline.

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Outline

• Problem Definition
• Notation
• The M/G/1 Queue with Server Vacations
• Extensions

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The M/G/1 Queue with Server Vacations

• Assume that the lengths of vacations are i.i.d.
  and are independent of the arrival process and of
  the service times of customers.

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The M/G/1 Queue with Server Vacations (contd)

• Remark 1
• pi(.)Pi(.)
• Remark 2
• Under a FIFO queuing discipline, the customers
  that a departing customer leaves behind are
  precisely those customers who arrived during the
  parting customers sojourn time.

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The M/G/1 Queue with Server Vacations (contd)
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The M/G/1 Queue with Server Vacations (contd)

• Primary customers
• Customers who arrive while the server is on
  vacation
• Secondary customers
• Customers who arrive while the server is busy
• Virtual 1-busy period
• The server begins service to a primary customer
  until the next time when that primary customer
  has departed and there are no secondary customers
  present in the system

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The M/G/1 Queue with Server Vacations (contd)

• Consider a random tagged customers as he
  departs from the system
• Primary customers

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The M/G/1 Queue with Server Vacations (contd)

• Secondary Customers
• Arrives since the current virtual 1-busy period
  begins
• Such virtual 1-busy period follows the same
  stochastic laws as in the system 2
• The number of customers the tagged customer
  leaves behind p2(.)

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The M/G/1 Queue with Server Vacations (contd)

• Total no. of customers the tagged customers left
  behind primary customers secondary customers

independent
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Outline

• Problem Definition
• Notation
• The M/G/1 Queue with Server Vacations
• Extensions

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Extensions

• To relax the assumptions for the following
  examples,
• Models where each vacation ends precisely when a
  fixed number of primary customers are waiting
• Cyclic queuing models in which the lengths of
  successive vacations are positively correlated.

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Notations

• Xn no. of primary customers present when the
  server returns from his nth vacation
• P(Xnk) P(Xk) for all n, k
• a(.)p.g.f of X

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Notations (contd)

• Consider a random primary customer.
• Xtotal no. of (primary) customers that arrive
  during current vacation.
• P(Xn) nP(Xn) / a(1)
• Ytotal no. of (primary) customers that arrive
  during current vacation but after the random
  primary customer arrived
• ß(.)p.g.f of Y
• P(Yk Xn) 1/n

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Proof

• To prove p1(z) ß(z) p2(z)
• No. of customers the sum of two independent
  random variables
• Find ß(z)

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Proof (contd)
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Conclusion

• The number of customers present in the system at
  a random point in time in equilibrium,p1(z)
• The number of Poisson arrivals during a time
  interval that is distributed as the equilibrium
  forward recurrence time (residual life) of a
  vacation,
• The number of customers present at a random point
  in time in equilibrium in the corresponding
  standard M/G/1 queuing system, p2(z)