 So, I will continue my discussion about the queuing system, you can say the features that are important in determining various aspects of a queuing system. So, L denoted our average number of people in the system, L queue was the average number of people in the queue. And then, we also had computed the average waiting time that a person will spend in the system, which includes waiting in the queue plus service time. And then, we will also talk about the waiting time in the queue, which we show also is important. So, anyway, so let me just continue the discussion about L and L queue. And so, this is as functions of rho, which is your lambda by mu, that is your lambda is the mean arrival rate and mu is the mean service rate. So, lambda by mu denotes your, in a sense utilization of the queuing system. So, now you can see from the figure, you can see from the figure that is on the screen, that as you know full utilization of the server, that means if rho is close to 1, then you see that your graph for L and L queue both are going to infinity, right. So, that means when lambda is close to mu, if rho is close to 1, that means lambda is close to mu, that is the mean arrival rate and the mean service rate are almost the same. In that case, you see the number of people in the system and the number of people in the queue, they will, the expected number will go to infinity, right. You can see the vertical, the approaching, the both the curves are approaching the vertical line. So, now this what we are saying is mean values, that means L is the average number of people in the system and L queue is the average number of people in the queue, which are going to infinity. So, it is not, see it is bad if once in a while your system has infinite people or very large people or the queue has becomes very large, but here it is saying that it is a mean behavior, that is on the average the system will have infinite people, very large people and the queue will also be very large, of course. So, that is not acceptable, finally, but anyway also what you want to say here is that now in this case, this does not happen in real life, because if the, if the big crowd, then it turns away lot of people. So, it is not that the queue and this will continue to grow. So, that does not really happen and therefore, what we are saying is that in real life, there will not be a balance, because the queue system, how many people turn away and so on. So, it will not follow the same pattern and therefore, when things become so bad, we cannot apply the same rules that we started with our assumptions and therefore, we will say that the system is not in balance. So, therefore, it cannot be measured or it cannot be analyzed by any of these models that we have written down. So, if you look at the mean length of 4 on the vertical line, then you draw horizontal line from 4, then you see that it will meet the corresponding utilization is 0.8. So, mean queue length, that means on the L Q curve. So, and so L Q is the first curve and then L is the second curve and similarly, if you look at the value of L Q equal to 5, the corresponding row value is 0.85. So, that means when you allow 4 people to wait in the queue, then the server will be ideal for 20 percent of the time, because row is 0.8. So, 0.2 is the fraction of time that the server will be idle and so 20 percent of the time the server is idle and if you allow the queue size to be 5, then it will be 15 percent idleness. So, you can see that there is definite conflict that is if you want that your server should not be idle for a very long time or even not then as we saw that the if the system is allowed to operate freely, then your queue and the people in the system and people in the queue will become very large. So, therefore, but then you do not want that to happen, because then you lose out on customers and so on and it becomes a chaos. So, there is a definite conflict between the issue, between the desire to obtain fully utilisation of a server and the desire to keep the mean queue length short. So, you can see that, so wherever there is goodwill of the customer is very important, certainly the persons who are offering the service would want to make sure that the queues do not become too big, but then where it is more important that means when the finances are important and you are having a service where they you know you cannot keep too many servers, because that means that many salaries and so on and then of course, your the servers will be idle for a long time. So, therefore, one has to balance. So, the system can of course, first of all this model has shown you that the system cannot be in balance if the arrival rate is equal to the service rate or even if the arrival rate is close to the service rate. So, the ideal ratio of arrival rate to service rate is something less than 1, that is acceptable, because your service rate must be more than the arrival rate otherwise and I mean that makes sense and so, but its specific value depends on the relative cost of idleness versus congestion. So, if for you it is very important that the system should not be congested, there should not be too many people in the system, then you will make sure that your service rate is much higher than the arrival rate and if you have limitations, you cannot provide you know more than one server, maybe let us say or the efficiency at service providing the service, then you know you will go for. So, idleness versus congestion. So, if you want the allow the server to be idle, you will not have that many people waiting in the system, because your service system will be such that your mu is much higher than the arrival rate, you will ensure that and in that case your value of LQ will be small, but if your mu is close to your lambda, then there will be congestion. So, one has to really strike a balance between you know idleness versus congestion. So, this is the whole idea. So, therefore, you see we can through these this model, we can study and decide what should be our level of service and given what is your level of you know customers arriving to the system. So, now even though we the little formula told us that your if you we have already computed average waiting time in the system and now we also know that WQ can be written as rho times W. That means the average number of people waiting in the queue is your utilization factor times the average waiting time in the system, but we will also like to compute this independently, because the distribution of WQ is also of importance and besides the WQ is of importance in hospital emergency rooms, because in a hospital emergency room the time that you are waiting to be taken to the doctor is important, because we want to cut that short. It is an emergency room and therefore, people need treatment fast. So, you would therefore, WQ is of importance and in hospital emergency rooms you would want to ensure that your WQ is not very large. And so, we want to look at it in a greater detail and also obtain the distribution of WQ. So, now of course, you can immediately write down this relationship, because if there is nobody in the system when the arrival comes for emergency treatment, then probability of the waiting time will be 0, because the patient will immediately be taken to a doctor for being treated. So, then in that case your waiting time will be 0. So, probability that WQ is 0 is P 0. That means, there is nobody in the system, which is 1 minus rho. So, now we will want to compute the distribution when a person coming to the system has to wait. That means, there is 1 or more than 1 person already in the emergency room and then the person has to wait. So, this we will compute try to obtain the distribution for WQ also. So, for n greater than 0 that means, if the any customers already present in the system, then the new arrival has to wait through an exponential service times until his or her own service begins. So, if you want to find out the probability that WQ is greater than T, where WQ is the waiting time in the Q. So, then probability WQ greater than T, so I will break it up into conditional probabilities and then add up. So, this will be probability WQ greater than T given that there are n customers and so we add up from 1 to infinity, because there has to be at least one customer to be being serviced, then only there will be a need to wait in the Q. So, this is n to 1 to infinity, probability WQ greater than T given that there are n customers in the system and this then would be written as. So, now when you write this probability then this will be P n into probability S n greater than T, because if these are the n service times then S n would be T 1 plus T 2 plus T n. So, this has to be greater than T, because the n people being serviced see you are in the you are waiting in the Q. So, then n services have to be completed and that takes because your waiting time is more than T. That means, these n services are taking more than time T and this into probability of there being n customers. So, then this will be, so I will write down for P n which is rho n into 1 minus rho and varying from 1 to infinity probability S n greater than T. Now, if I take rho outside and then I write this as rho, so this will be rho n minus 1 into 1 minus rho and this is probability S n greater than T. Now, see at n equal to 1 the value here is rho into 1 minus rho and this is probability S 1 greater than T. So, if I rewrite this as rho outside and now at n equal to 0 your P n would be just 1 minus rho, because rho n rho raise to 0 would be 1. So, this would be 1 minus rho, so same thing and here it will be S 1 greater than T. So, the same event is being written here and therefore, all subsequent ones will be the same. So, this helps you to because now you have this summation, the probability sign is missing here. So, please write this will be P in probability of S n plus 1 greater than T, I will just write it, there should have been probability. So, probability S n plus 1 greater than T, so therefore, so now you have, because the summation is from 0 to infinity, so then we can sum it up easily. And therefore, you can you know just expand this and so here this is no people, then the service time of the rho into 1 minus rho actually would be the you know 1 percent in the system, then the waiting time, this is the service time of the first person is of the one person already present in the system is more than T and so on. So, when you take out rho then this is what you can, you just expanding this expression like this, this series. And this now we know, we recognize this because when we computed the distribution for the average time in the system. So, then this was, this is nothing but and that is why we wanted the sum from 0 to infinity. So, this is probability W greater than T and we already know the distribution of W greater than T. So, we know this probability, so therefore, this is rho into e raise to minus mu 1 minus rho T. So, we have computed this probability W q greater than T, which is equal to this. Now, some of the rho is creating the problem, because I cannot call this exponential distribution, why? Because if you write the pdf of W q then this will be you know this you will write as 1 minus probability W q less greater than T. This whole thing, so then d dt of that will give you the pdf of W q. So, this will be minus rho into, because this is 0 and I will write down the, so this one is rho into e raise to minus mu 1 minus rho T substitute for this here, then d dt of this rho is outside. So, then differentiation gives me this expression, which is equal to this. So, therefore, this would not be an exponential distribution, because exponential distribution would be this rho is the extra part. But if you consider the conditional distribution, so now that is important conditional distribution of W q given that W q is greater than 0 it does have an exponential distribution, because now you will write the conditional part and this will be equal to probability W q greater than T, because when you take the intersection obviously T is positive, so W q greater than T. So, the intersection of these two becomes just this event and divided by probability W q greater than 0. See, remember this you have to write as probability W q greater than T intersection W q greater than 0 and then it will be probability W q greater than 0. This is what our formula for conditional probability is, but this is same as this of course, given that T is positive. So, this event is equivalent to this event given that T is positive and then you divide by probability W q greater than 0. So, therefore, this becomes now rho e raise to minus mu 1 minus rho T divided by rho, because probability W q greater than 0 is this is what we computed here W q is not 0. So, remember our W q probability W q equal to 0 was 1 minus rho, because if the person does not have to wait that that thing there is no person in the system and so therefore, P 0 this probability is equal to P 0 which is 1 minus rho. So, if you want probability W q greater than 0 then it will be 1 minus of 1 minus rho which is equal to rho. So, probability W q greater than 0 is rho, so we divide this. Now, the rho cancels out and this is e raise to minus mu 1 minus rho T. So, this now is coming from that means, this now represents because the conditional part W q greater than T given that W q is positive. So, this is this which matches with our exponential distribution and so the conditional distribution of W q is exponential with parameter mu into 1 minus rho T. So, yeah so this is therefore, I again just want to repeat that this is not the conditional just remove this. So, it is just probability W q greater than T is that what you are computing you started writing out is this and then this is I just you know rewrote this expression and just so that I can relate this series with the series probability W greater than T that was helpful and then we saw that the distribution is not matching with the exponential. So, it is only when you take the conditional probability that you will get fine and so this already we had seen and therefore, this is rho. So, that is it. So, therefore, and this will be useful at times you may really want to know the distribution of the. So, this will be conditional distribution of W q which you can recognize otherwise the distribution of W q is given by this I mean the f q t f W q t. Now, let us compute the expected time or the average time spent in the q waiting for to be serviced. So, that we will now compute independently because this is only the conditional part. So, you will say that the expected time in the system is equal to the expected. So, the time in the system is time in the q plus service time. So, expected value of that. So, expected time in the system is expected value of time in the q plus the service time right and so this can be written as expected time of time in the q plus expected service time. So, this if we denote by W q. So, this is the convention we have been following that the variable is also treated as the denoting the expected value of that variable. So, this is expected time in the q plus expected service time is 1 by mu right because your number of services is with parameter mu exponential with parameter mu. So, 1 by mu is the expected service time. So, this is what you have and therefore, W q is W minus 1 by mu and you can. So, we know W expected value of W which is 1 upon mu minus lambda minus 1 by mu. So, this becomes what we had computed rho times W. So, W q is equal to rho times W and which we have already seen through using the little formula. So, we saw that the conditional distribution of W q and W are both same exponential and the parameter was yeah I mean the same parameter. So, they and now also I want to make an observation that the little formula that we obtained were under special conditions, but it turns out. So, that means the relationship between L and L q W and W q and L and W. So, they are all in fact what it means is that if you can find any one of the quantities L L L q W W q then you can find all the other three. So, all the four are related and it turns out fortunately that under very general conditions these formulae are valid. So, therefore, you know computing any one of them would help you to get the values of the others. Now, I want to continue this discussion on these quantities L L q W and then also show you how we then through these analyze these weighting weighting systems or with a qing model. So, this particular example is a small one, but anyway this is some Sheldon Ross and what it says is that machinery in a factory break down at an exponential rate of 6 per hour. So, that means the arrival of the machinery for repair is 6 per hour and then there is a single repair man who fixes machines at an exponential rate of 8 per hour. So, the arrival and the service rate is all provided to you. The cost incurred in lost production when machines are out of service. See the machines come for repair and they are waiting. So, the repair man has to repair them. So, while the machines are out of service they incur a cost because there is lost production. So, rupees 100 per hour per machine is lost to the organization. So, the cost of lost production is rupees 100 per hour per machine. This is what is given to us. Now, we want to find out what is the average cost rate incurred due to failed machines. So, while this is machines are waiting to be repaired, they are not producing and therefore, there is a loss to the organization and this is the rate. So, you want to find out the average cost rate. Now, the average cost rate will be dependent on the number of machines which are in the system which are either waiting to be repaired or which are being repaired. So, that will be our number L. So, therefore, you see L lambda is 6, mu is 8. So, therefore, average cost rate we will write as rupees 100 into average number of broken machines which is either waiting to be repaired or they are waiting or they are being repaired. So, this will be rupees 100 into L. L is the average number of broken machines which are in the system. L gives you the average number of people or customers in the system. So, therefore, this is rupees 100 into lambda upon mu minus lambda which is 100 into 6 upon 8 minus 6. So, rupees 300 per hour. So, the loss. So, therefore, this is a very important parameter because the system would now like to evaluate whether the repairman that they have is good enough or they need to have more repairman because it depends on what is the how much the loss to the system compared to the salary of a repairman and so on. So, that question you see will always be running through all these examples and you want to analyze this. So, this should hopefully help you in your decision process. Let us take another example. So, you have now a pump station, a single pump station. So, a single pump petrol station. So, there is only one pump and so cars that come for taking the petrol have to wait in the queue. If one is being serviced, once the serviced car is done with then the other will come from which is waiting in the queue. Now, inter arrival times are exponential with mean 12 minutes and that means the inter arrival time the average time between two arrivals is 12 minutes and the service time is exponential again with mean service time 6 minutes. So, average time it takes to fill up a car is 6 minutes and waiting space is unlimited. So, I have not put any question here, but as we go along we will see what are the kind of questions we want to answer here. So, let us see lambda therefore that means the arrival rate is 5 per hour because arrival time inter arrival time with mean is 12. So, therefore the number of arrivals per hour is 5 and similarly the service rate mu will be 1 by 6 into 60 because this is per minute 6 this is this is all in minutes. So, you convert to hour. So, this will be 10 per hour that means the service rate you can on the average you can fill up 10 cars in an hour. So, rho the utilization factor or the traffic intensity we have lot of names for this. So, rho is 5 by 10 5 is your arrival rate and 10 is your service. So, 5 by 10 therefore this is 0.5. So, the traffic intensity is not very high 0.5 is not considered to be very at means the petrol station is not very busy right now with the kind of service rate and the kind of arrival rate. Then if you want to look at the probability that there is no car at the petrol station. So, that will be 0.5 1 minus rho which is 0.5 again which is a high probability then P n is the n n cars at the station. So, that will be 0.5 into 0.5 raise to n and then the mean number of cars at the station mean number of cars would be lambda upon mu minus lambda or well this is why am I writing this as how in fact this is sorry this is rho upon 1 minus rho. So, the mean number of cars at the station is 0.5 upon 1 minus 0.5 which is 1 and l q the car waiting in the q is 0.5. So, this is the idea. So, now we want to analyze the system through these quantities that we have computed and you know like you want to again answer the question that in case the arrival rate increases then supposedly the traffic intensity will go up because this number will go up and then what kind of numbers l and l q will be there the values will also go up and therefore, the petrol or the station owner may want to ask a question as to should we install a faster pump and so on or of course, more than one pump you will the system that model we will discuss later on when you have more than one server right now we are talking about only one server systems and therefore, the only option that the man may have in case the arrival rate goes up in the other option would be to install a pump which is you know filling up cars at a higher rate. So, we will just look at the analysis with the numbers. So, through this example which we are just discussing I again want to raise the issues about you know validity of a model. So, it is very important that you keep recalling what are the assumptions under which we are working and what are the. So, for example here I wrote that waiting space is unlimited, but you know that in a petrol station waiting space cannot be unlimited and usually you have space for 2 to 3 cars. So, but of course, the given data right now it did not really matter because your l was 1 and your l q was quite small right l q was 0.5 I think l q is 0.5. So, therefore, that right now it is not an issue whether the space is limited or unlimited, but in case your data changes then to say that your waiting space is unlimited is not a very valid assumption. So, therefore, one should always keep this in mind that the better model would be when you talk of queues with limited capacity or with limited waiting time. So, that would be a better model for such an example then the arrival pattern is state dependent. Now, here to assume that lambda will remain the same all the time is not correct because if there you know already 2 to 3 cars waiting the person may want to go to the next petrol station. So, therefore, this is not and arrival process is not stationary also. So, it is state dependent and also not stationary because during the rush hour there may be the lambda may be higher then corresponding to when it is a so therefore, the lambda itself may change during the day. So, the lambda is not stationary and it is also a state dependent because people do not like to wait for too long because as you know you can always drive fast you know further and get another petrol station. There may be other considerations also that is true, but sometimes people like to wait at a particular station because they are familiar with it people know them and they like and there are so many other reasons. Then also we must keep this in mind that whatever computations we are doing remember they are they are not giving us accurate information about L and L Q and the remaining other parameters W and W Q, but we can certainly change the values of lambda and mu and of the system. So, that means we can study the changes in the system and then correspondingly see what are the changes in these numbers L, L Q, W and W Q. So, this model certainly will help you to study the changes whatever changes take place in the system then we can accordingly find out the changes in L and L Q. So, that is what I want to so that is what I have stated here that the real use of the model is in evaluating the effect of changes in lambda and mu. So, for example, the station owner has the choice of has the alternative of installing a faster pump. If he does that then the mean service time is reduced to 4 minutes per car. It was earlier this was earlier 6 minutes per hour on the average. So, now the average time has gone down to 4 minutes. That means the patrol pump can service 15 cars per hour. So, by installing a faster pump the mean service time is reduced to 4 minutes that is the pump can fill up 15 cars per hour. So, your row would now become 5 upon 15 because the arrival rate is 5 per hour. So, the intensity traffic in sent is the intensity or as you call it the utilization of the patrol pump and so on. So, that is 5 by 15 which is 1 by 3 and this is 0.33. So, this is less than 0.5. So, the earlier one was 0.5 traffic intensity. Now, it has come down to 0.33 and probability that is no car at the station. So, that comes out to be 1 minus 0.33 because it will be 1 minus row and so that is 0.67 which is greater than 0.5. So, that means the patrol pump would be vacant for more time. The fraction of time will be higher here than here because there it was 0.5. Since row is 0.5 so 1 minus row is also 0.5 and your L the average number of people in the system or at the patrol pump that means the number of cars getting filled up or waiting to be filled up. So, that will come out to be 0.33 by 0.67 which is also less than 0.5. So, therefore, this does not warrant installing a faster pump because your patrol pump is vacant longer the intensity traffic intensity has come down and so on. So, if you are looking at it from the view point of the patrol pump owner then certainly it does not warrant installing a faster pump. In case the arrival rate goes up that means you have 6 cars per hour instead of 5 cars per hour and with the current pump that you let the man has then this will be row will be 6 by 10. So, this will be 0.6 that means your traffic intensity will go up from 0.5 to 0.6. Your probability of there being no car in the system is 0.4 and your L is 1.5 and your L Q is 0.9. So, therefore, you see that is what I am saying that now with the model you can play around and for different values of lambda you can figure out what is how these numbers are changing and then if we can as I said you know the losing the good will of the customer versus the cost of installing a faster pump and so on. One can the owner can study all those things through this model. And therefore, that the basic contribution of this model lies in being able to study the various changes that will take place in your you know you can call them parameters when you change your lambda and new change. So, now I will after having discussed the one server model we will now look at the situation when there is more than one server. So, this is the model would be MMS that the same the pattern arrival pattern and the service pattern are the same. You can say that and therefore, the number of servers is now more than 1. So, in that case you see as long as there are number of people is less than s then your service rate will be because whoever is there in the system if there are n people in the system all will be serviced. And therefore, your service rate will become n times new that is understandable because everybody is being serviced. And therefore, the n people who are being serviced simultaneously. So, the service rate you can say has gone up to n mu and if you have more than n s people in the system then of course, it will remain at s mu because only s people can be serviced since you have s servers. So, therefore, then the service rate will be s mu, but I will try to explain. And therefore, diagrammatically if you look at the transition diagram here then you see arrival rate is the same lambda which is you know. So, lambda is the arrival rate, but the service rate changes. So, if you have 1 person in the system then it will be mu because you serviced and therefore, you get back to 0 state. If you have 2 people then 2 are being serviced simultaneously and therefore, you have the rate at which the system can transition from 2 to 1 will be 2 mu. So, this should be understood very well because so what we are saying is that many people are being serviced and therefore, the probability of transitioning that means, the rate at which you can transition from 2 to 1 will be 2 mu. And similarly, with 3 people the rate of transition will be 3 mu, but the arrival pattern is the same and therefore, the arrival rate is lambda. So, this will go on up to s minus 2 and then up to s minus 1 and when you have s people in the system if your state is the system is occupying state s, then it will be s mu and thereafter the service rate will remain at s mu. The arrival pattern would be at the rate of lambda, but the service rate would then remain at s mu. So, this is the idea and so here is the service rate depends on the number of people in the system that is if the number of people is less than s then it is n mu and if it is more than s then it is s mu. Now, let us understand the assumptions which is very important and what we are assuming is that all operators operate at the same mean rate. So, right now this is a simplification because obviously, it will get very complicated if I have different servers with different service rates. So, therefore, we are assuming that all operators operate at the same mean rate mu. Therefore, I am saying that when there are n people in the system the service rate will be n mu and when they are more than n it will be s mu. So, this is possible only if I make the assumption that all operators or all service people are operating they have the same mean rate mu. So, this makes it because then we do not have to keep track of which servers are busy you know because then we will have to accordingly keep changing the service rate and that will become quite problematic right. And then again I feel that this is not very this this can be treated as a realistic assumption because you know a person may be a little more efficient than the other but the differences cannot be very very large to really take care of them in the system right in the model. So, this is what one assumption and then the second assumption which is important is that departures will be one at a time that is probability of service of two or more services being completed exactly at the same time is 0. So, this probability that means the departures because that is important of the you know when we say that we are looking at M M S system and yes I will have occasion to explain you know through the when we talk of Markov processes we will discuss in detail. And so anyway and when we are talking of Poisson process remember I had told you that there is a always a small interval there is a small enough interval in which we say that the probability of one arrival is probably you know there is something like lambda delta t right and then for more than more than one arrival it was of the order delta t square and higher order right. So, therefore we were neglecting the so the probability was again that means we assume that exactly at the same time two or more customers will not leave the system. So, the service will not get completed exactly at the same time for more than one people. So, therefore there will be a distinct interval between two departures. So, therefore we can model this as a birth death process because our birth and death process the assumption when we talking of M M S. So, then the basic assumption is that your arrivals and departures are distinct at distinct times you cannot have more than one departure or one arrival at the same time. So, once we make that assumption and of course the second assumption is that the operators are all operating at the same efficiency. So, the mean rate is same then we can process this system as a birth death process right and so now one can write down the balance equations. So, this is the first one which is easy to understand right because you have you already have one person then the rate at which it can depart is mu. So, mu p 1 must be lambda p naught one person can arrive when you have 0 when you are in a 0 state. So, then this and this gives you this. Similarly, when you want to write for so two people in the system then it will be lambda p naught you can go to p 1 sorry to 1 and from here when you have two people then at the rate 2 mu you can go to again because one departure one person gets serviced and so you again go to p 1 and here it will be lambda plus mu right p 1. That is why the transition diagram and even when I were discussing mm 1 system I had explained to you how you can you know interpret the transition diagram. So, it will be lambda plus mu. So, it is actually not there is nothing new here except that you have to remember that. So, I have not written down the remaining things because it is understood that up to s minus 2 you will have this thing and then after that you will the moment you have s people in the system s or more then this is the balance equation that you will get right. So, once you have written down these balance equations immediately you can start solving. So, p 1 in terms of p naught will be lambda by mu p naught and then if in the second one if you substitute for p 1 in terms of p naught then you get your 2 mu p 2 as. So, the expression 2 mu p 2 simplifies to equal to lambda square upon mu p naught and therefore, p 2 is equal to lambda square upon 2 mu square p naught. So, note the correction that 2 was missing. So, it should be lambda square upon 2 mu square p naught. So, all the probabilities can be computed in terms of p naught. So, important thing is that the moment you have more than one server things change a little and one has to understand under what assumptions you are now modeling the situation and so I have tried to explain to you how we will under what assumptions we will treat this as a birth death process. So, the services are all at the same rate that means all servers have the same efficiency and that no more than one departure at exactly at the same time. So, there will be distinct interval of time between any two departures from the system. So, under this you can easily write and then of course, the service rate changes depending on the number of servers you have and so under all these three assumptions you can write these balance equations and then you will try to get the probabilities in general formula and then we will again compute your quantities L, L, Q, W, W, Q to get an idea about. So, therefore, your traffic intensity also will change. You can see that because the service rate is changing and therefore, your traffic intensity will also change. So, all these again open up a very interesting situation and we would like to look at them.