**CHAPTER 6 6.5. NUMERICAL RESULTS**

**6.5 Numerical results**

To observe the efficiency of the proposed model, we perform some numerical experiments in this section. The performance measures observed are the availability of the server and the average queue length. Mainly, we want to depict the influence of retrial on the WV

*CHAPTER 6* *6.5. NUMERICAL RESULTS*

system. We plot the change in the performance measures for different retrial rates along with the change in the other parameter values. For simplicity, we first assume that the service time distribution is exponential. Later, we compare the effect of different service distributions on the model.

### Effect on system availability

In Figure 6.1, the plot of system availability against vacation service rate is shown. We
have taken *a* = 0*.*7*, λ* = 1*.*85*, µ*_{b} = 2 and *θ* = 0*.*3. When *µ*_{v} = 0, the system does not
serve during the vacation period and our model boils down to a pure vacation model.

For *µ*_{v} =*µ*_{b} = 2, the server serves the customer at the same rate throughout and so this
becomes equivalent to a system where the server never takes vacations. The graph gives
the bridge between a pure vacation retrial model and a non-vacation one. Within this
range of *µ*_{v}, the system availability can increase by upto 60%. Another observation is
that if there are no retrials (*α* = 0) the availability of the system increases by upto 50%

with the increase in vacation service rate. The number of retrials (per unit time) increases
the system availability, but the system almost becomes unaffected by retrial rate beyond
*α* = 10. This is because, when the mean retrial time _{α}^{1} is too small, this is similar to a
system without retrials.

Figure 6.2 and Figure 6.3 are plots of system availability against the retrial rate for
different vacation duration rates*θ*. The retrial rate increases the availability of the server.

This increase is monotone for smaller rates of*θ*(in Figure 6.2). But this behavior changes
as the vacation duration rate increases beyond*θ* = 1*.*8. For *θ*= 8, the system availability
remains almost stable with respect to the retrial rates as we can see in the graph (Figure
6.3). So the retrial rate affects the system availability if the vacation duration rate is
below 8, *i.e.,* for smaller durations of vacations.

The blocking probability is the probability that the system is not available *i.e.*,*P*_{B} =
1*−P*_{A}. Therefore the retrial effect on the blocking probability will be the reverse of the
effect on system availability.

*CHAPTER 6* *6.5. NUMERICAL RESULTS*

### Effect on average queue length

We compare the mean queue lengths of this system for three different values of vacation duration rates in Figure 6.4. The retrial rate can decrease the mean queue length. Here the vacation duration rate plays a role to minimize the average queue length even further.

Figure 6.5 shows that the increase in retrial rate can enhance the efficiency of the model by decreasing the mean queue length. The value of queue length, or equivalently the value of waiting time in queue, can be reduced if the retrial rates as well as the vacation duration rates are increased.

The mean waiting time is the ratio of mean queue length over the arrival rate*λ*. Figure
6.6 gives the mean queue length of our system without retrial. When mean retrial tends
to*∞*, our model gives almost the same result as in Lui [113].

### Effect of service distribution

To see the effect of service distribution on the model, we have plotted in Figure 6.7 the system availability for an exponentially distributed service time distribution and for a Erlang-2 distributed service times having the same mean. The parameter values are given in the graph. We have taken the same mean values for both the distributions. The graph shows a vast difference in server availability if the service distribution is changed. Server availability for the exponential distributed service model is about 40% more than the one with Erlang-2. Therefore the choice of service distribution is extremely important in order to enhance the efficiency of the model.

*CHAPTER 6* *6.5. NUMERICAL RESULTS*

0 0.5 1 1.5 2

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

service rate µ_{v}

Probability of server availability

α=0 α=1 α=10

Figure 6.1: Probability of server availability vs vacation-service rate with *a* = 0*.*7, *λ* =
1*.*85, *µ*_{b} = 2, *θ*= 0*.*3.

0 2 4 6 8 10

0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.62

retrial rate α

Probability of server availability

θ=0.001 θ=0.5 θ=1

Figure 6.2: Probability of server availability vs retrial rate with*a*= 0*.*7,*λ*= 1*.*85,*µ**b* = 2,
*µ*_{v} = 1*.*8.

*CHAPTER 6* *6.5. NUMERICAL RESULTS*

0 2 4 6 8 10

0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.52

retrial rate α

Probability of server availability

θ=1 θ=1.8 θ=3 θ=8

Figure 6.3: Probability of server availability vs retrial rate with*a*= 0*.*7,*λ*= 1*.*85,*µ*_{b} = 2,
*µ*_{v} = 1*.*8.

1 2 3 4 5 6 7 8 9 10

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

retrial rate α Mean Queue length L s

θ=.5 θ=1 θ=1.5

Figure 6.4: Mean queue length vs retrial rate with *a*= 0*.*7, *λ*= 1*.*85, *µ*_{b} = 2, *µ*_{v} = 1*.*8.

*CHAPTER 6* *6.5. NUMERICAL RESULTS*

0 0.5 1 1.5 2

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

service rate µ_{v}
Mean Queue length L s

α=0 α=.5 α=1

Figure 6.5: Mean queue length vs vacation-service rate with *a* = 0*.*7, *λ* = 1*.*85, *µ**b* = 2,
*θ* = 1*.*8.

0 0.5 1 1.5 2

1 1.5 2 2.5 3 3.5 4 4.5

service rate µ_{v}
Mean Queue length L s

θ=.5 θ=1 θ=1.5

Figure 6.6: Mean queue length vs vacation-service rate with *a* = 0*.*7, *λ* = 1*.*85, *µ*_{b} = 2,
*α*= 100.

*CHAPTER 6* *6.5. NUMERICAL RESULTS*

0 2 4 6 8 10

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

retrial rate α

Probability of server availability

θ=.5, Exp θ=.5, Erlang 2

Figure 6.7: Probability of server availability vs retrial rate with*a*= 0*.*7,*λ*= 1*.*85,*µ**b* = 2,
*µ*_{v} = 1*.*8.

## Chapter 7

## A Queue with Impatient Customers

In a communication network the jobs which do not get service upon arrival wait in a queue. The jobs can become impatient due to high waiting times or due to uncertainty of receiving services and may leave the system unserved. This scenario often occurs in Optical Burst Switching (OBS) networks. OBS is a technology for reducing the gap between transmissions and switching speeds. In OBS, incoming traffic from clients at the edge of the network is aggregated at the ingress of the network according to a particular parameter. This is then assembled and is further transmitted through WDM links. The operation of an OBS controller can be seen as a queue with reneging or impatience [30].

In OBS networks, a control packet is sent first, on a separate signaling channel, to set up a connection. It is followed by a data burst without waiting for an acknowledgement for path establishment [131]. In particular, when a path is not assigned, the burst control packet is accepted to the queue and is kept waiting for a path. If its delay budget is lower than the effective processing delay, the packet becomes impatient and leaves the system unserved. To have more efficient use of the network, the loss of packets has to be reduced for better performance of the network. Therefore, study on impatience of a request in WDM links is essential for the benefit of better system performance. This chapter presents a detailed study of stationary distribution of packet waiting times, the queue length distribution and the mean number of packets served (which do not abandon