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Unformatted text preview: em. Assuming that ∏ > µ1 and ∏ > µ2 , the
departures from each queue are Poisson of rate ∏. Let X(t) be the state of queueing system 1 and Y (t) be the state of queueing system 2.
Since X (t) at time t is independent of the departures from system 1 prior to t, X (t) is
independent of the arrivals to system 2 prior to time t. Since Y (t) depends only on the
arrivals to system 2 prior to t and on the service times that have been completed prior to
t, we see that X (t) is independent of Y (t). This leaves a slight nitpicking question about
what happens at the instant of a departure from system 1. We have considered the state
X (t) at the instant of a departure to be the number of customers remaining in system 1
not counting the departing customer. Also the state Y (t) is the state in system 2 including
the new arrival at instant t. The state X (t) then is independent of the departures up to
and including t, so that X (t) and Y (t) are still independent.
Next assume that both systems use FCFS service. Consider a customer that leaves system
1 at time t. The time at which that customer arrived at system 1, and thus the waiting
time in system 1 for that customer, is independent of the departures prior to t. This
means that the state of system 2 immediately before the given customer arrives at time t is
independent of the time the customer spent in system 1. It therefore follows that the time
that the customer spends in system 2 is independent of the time spent in system 1. Thus
the total system time that a customer spends in both system 1 and system 2 is the sum of
two independent random variables.
This same argument can be applied to more than 2 queueing systems in tandem. It can also
be applied to more general networks of queues, each with single servers with exponentially
distributed service times. The restriction here is that there can not be any cycle of queueing
systems where departures from each queue in the cycle can enter the next queue in the cycle.
The problem posed by such cycles can be seen easily in the following example of a single
queueing system with feedback (see Figure 6.11). 258 CHAPTER 6. MARKOV PROCESSES WITH COUNTABLE STATE SPACES M/M/1
∏
✻ Q ✲
µ ∏
✲ 1−Q Figure 6.11: A queue with feedback. Assuming that µ > ∏/Q, the exogenous output
is Poisson of rate ∏. We assume that the queueing system in Figure 6.11 has a single server with IID exponentially distributed service times that are independent of arrival times. The exogenous arrivals
from outside the system are Poisson with rate ∏. With probability Q, the departures from
the queue leave the entire system, and, alternatively, with probability 1 − Q, they return
instantaneously to the input of the queue. Successive choices between leaving the system
and returning to the input are IID and independent of exogenous arrivals and of service
times. Figure 6.12 shows a sample function of the arrivals and departures in the case in
which the service rate µ is very much greater than the exogenous arrival rate ∏. Each
exogenous arrival spawns a geometrically distributed set of departures and simultaneous
reentries. Thus the overall arrival process to the queue, counting both exogenous arrivals
and feedback from the output, is not Poisson. Note, however, that if we look at the Markov
process description, the departures that are fed back to the input correspond to self loops
from one state to itself. Thus the Markov process is the same as one without the self loops
with a service rate equal to µQ. Thus, from Burke’s theorem, the exogenous departures are
Poisson with rate ∏. Also the steady state distribution of X (t) is P {X (t) = i} = (1 − ρ)ρi
where ρ = ∏/(µQ) (assuming, of course, that ρ < 1).
exogenous
arrival reen
✄ ° tries
✄° ❄❄
❄
✂✁ ❄
❄
✂✁ endogenous
departures ❄
❄ endogenous
departures ✄°
❄
❄
✂✁ ❄ Figure 6.12: Sample path of arrivals and departures for queue with feedback.
The tandem queueing system of Figure 6.10 can also be regarded as a combined Markov
process in which the state at time t is the pair (X (t), Y (t)). The transitions in this
process correspond to, ﬁrst, exogenous arrivals in which X (t) increases, second, exogenous
departures in which Y (t) decreases, and third, transfers from system 1 to system 2 in which
X (t) decreases and Y (t) simultaneously increases. The combined process is not reversible
since there is no transition in which X (t) increases and Y (t) simultaneously decreases. In
the next section, we show how to analyze these combined Markov processes for more general 6.7. JACKSON NETWORKS 259 networks of queues. 6.7 Jackson networks In many queueing situations, a customer has to wait in a number of diﬀerent queues before
completing the desired transaction and leaving the system. For example, when we go to
the registry of motor vehicles to get a driver’s license, we must wait in one queue to have
the application processed, in another queue to pay for the license, and in yet a third queue
to obtain a photograph for the license. In...
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.
 Spring '09
 R.Srikant

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