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Queueing Theory



Waiting time is never a preferable option for any of us. In order to diminish that disgust cost is to be incurred. For both the reduction and avoidance of such time costs investments. When deciding that wither we should invest or not the cost in terms of waiting time is compared with the cost that is incurred to reduce the waiting time. The cost that is overall lower is usually opted and is considered as an alternative that can be worked upon. For the purpose of this comparison some techniques and models are used. These models and techniques are usually used in terms of calculating the costs of waiting time. One of the best model that stands at the top of the list is the queuing model (Willig, 1999).


Queuing is primarily considered an extension of Applied Probability theory. It has range of applications and it used in different realms, such as communication, networks, computer systems, machine plants, supermarkets and call centers.


Queuing theory can be explained or elaborated as follows. Think of a service center and the populace, which visits Service Center, frequently, to acquire services. Because of the operational capacity, Service Center can only serve a certain numbers, from the populace. When a Service Center is burdened, a fresh client has to wait, until system becomes operational or available again. This helps us acknowledge three elements or aspects of service center; 1) Population, 2) Service facility and 3) Waiting line. Queuing theory helps in forming network of Service Centers, which can avoid waiting lines and can keep Service center operational for clients, from a certain population.


An airline counter could be an opposite example for a Service Center, where travelers are presumed to check in. Most of the time, check-in services are provided by a single employee. In case there are number of travelers and only a single employee to provide services, passenger or traveler has to wait in the queue. This is referred as FIFO service, First In, First Out.

Examples, pertaining to Queuing theory, from the realm of Network are the buffers in Dimensioning of routers, computing and the determination of the trunks’ digits, in the Head Quarters, which is located in POTS, Comprehensive or end achievingcomputation, from network systems. 


Queuing theory addresses or answers questions, such as the mean time client splurges waiting in the queue, the mean time system takes to respond, mean consumption of a service feature or facility, mean distribution of clients (in a queue), mean distributing pertaining to clients. These core questions are scrutinized in stochastic situation, in which client arrival and response system, of Service Center are taken randomly.


A client comes to a Service Center, in a random manner. There can be one or more than one servers, provided by Service facility, with the capacity or ability to serve only a single client or customer at a given time. The required time, to provide services to a customer, are also assumed or modeled random. Following figure shows an example of that kind of queue.An example is shown in the following figure of that kind of queue.






There are some assumptions to be considered:


The population size has been considered infinite, Cnis the last customer has arrived at the time. It has been considered gap or interval time, amid two clients and it is defined as

We have also presumed tn. which represents arbitrary times, are random variables that means that they are autonomous or independent from one another. In addition, all the tn has been selected from single distribution, with the help of distribution function.


And the density of probability function:


  • We have also considered Xn, service times for every customer Cn, a random variable. Both Xn and Cn are random arbitrary variables, which have single/common distribution function, B (t) and single/common probability density function b (t).


There is high probability that Queuing systems can be unique not just because of dissimilarity in distributions of inter-arrival and service responses, but also because of number of servers, of each Queuing system, and length of waiting line. In addition, Queuing system could be unique because of service discipline. Some of the most common disciplines of service are:


FIFO: (First in, First out): When client joins the end of a queue, when service facility is overburdened, by clients or work.

LIFO: (Last in, First out): In a scenario, where service facility or service center is burdened, a client would advance towards the head of a queue. A client would be addressed or served immediately, if no other client arrived.

Random Service: With such service method, clients are dealt or served arbitrarily.

Round Robin: A certain time is allocated to each client. If during this time, required services are not comprehensively provided, client re-enters the queue.

Priority Disciplines: Clients are served or addressed as per their priority.


For smaller description, of queuing systems, commonly Kendall Notation is used.  A queuing system characterization appears as A/B/m/N-


In the description, A symbolizes or represents distribution of interval time, B represents service time’s distribution, m represents figure of servers, and N represents the upper limit of waiting line, in a finite case. S, which is an optional, represents employed service discipline. The default service discipline is FIFO, which means if we omit S then FIFO will be the presumed discipline. For B and A the following symbols are very common:



  • M (Markov): M denotes the distribution exponentially withand, where > 0 is a measure. M represents the property that the exponentially distributed variableremains only distribution that is distribution of continuous nature with Markov property, i.e. it is memory less.


< >D (Deterministic): all values from a “distribution”  deterministic in nature are constant, i.e. constitutes ofsimilar values valueEk (Erlang-k): Erlang Distribution having k stages (k 1). For the Erlang-k distribution we have



Parameter is >0. This sort of distribution is commonly used for modeling phone-call arrivals, in Head Quarters (Rasing, 1990). The subsequent graph comprehensively depicts density of Erlang-k distribution with mean 1 and various k values


< >Hk (Hyper-k): Hyper distributed exponentiallyhaving k stages. So we get we have


< >G (General): general distribution, not further specified. In most cases at least the mean and the variance are known.MARKOVIAN SYSTEMSM/M/1 Queue:Number of clients, in a system, is immense. The effect of single customer, on a system, is minimal; a solitary client makes only make a small percentage of system resources. Each client is unique or independent

The M/M/1-Queue has exponentially distributed interval times. It has  as measure, of a distribution of exponential nature, whereas denotes the measure of exponential distribution of service times. This system only has one server; the discipline system it uses is FIFO. The length, of a waiting line, is infinite. The M/M/1-Queue is considered a typical death/birth system, where only single event occurs, at a given time. This event could be the arrival of new customer/client or this could be completion of services, for a certain client. What constitutes M/M/1 system to be simple is rate of arrival and the rate of service, which are not dependent on state.

Utilization =


Average of customers in the queue is denoted by:

Mean time of response: (from Little Law)

The chance of that the system has k or more than k customers is denoted by:

Following graph shows mean for customers vs. factor of utilization.

The graph for mean delay vs. utilization:



< >M/M/m-Queue




The mean number of customers in the system is given by:

Due to the absence of any servers to service the customers it must be evaluated that every arriving customer would enter the queue. This is usuallyobserved in calls and signifies the chance that a latest incoming call at call center will get no chunk, given that the arrival times and the difference between them along with the calltimes are distributionsof exponential nature. There probability can be computed as:

This formulation denoted as “Erlang C formula”.

Following graph is mean number of customers in a system that for m=20.

< >The M/M/1/K-Queue:. The FIFO-order would be utilized for serving the customers, there would be single server but systems would hold up K customers only. As there is K customers present in the system but when a new customer arrives and would be considered as lost as it would be dropped from the system and never reached again routinely called as blocking. This particular behavior is much needed as the customer does not have to wait for the vacant place until it’s done; therefore the process would not be Markovian. The fullknowledge regarding the state of the system is denoted by the customers of the system in M/M/1 case. This system is also called as pure death-birth system and is best-suited for real systems likewise routers where space of buffers is finite/ quantifiable(Teorisi, SUCU, & Simdekler, 2003). 

The average number of systems’ customers is denoted by:

The graph below is average of customers of the M/M1/10 queue

The loss probability is simply the probability that an arriving customer finds the system full, i.e. the loss probability is given as PK with

The graph showing probability of loss for the M/M/1/20 queue:

< >Comparison of these three systems:m, M/M/m systems as well as m queues systems of M/M/1 type along with the rate of service   are parallel in nature. In such a scenario, every client enters into systems having same probabilities. The results could be seen in the following diagram:-

The results of the second solution provide the effective results smaller delays, which is followed by the initial solutions. The last solution is considered as the worst one. The initial systems are related with the M/M/m systems, where the entity serving has the arrival time  and    service rates. The second system is correlated with the M/M/1 system having arrival rate having rate of service. From single user view, the last component corresponding with M/M/1 system having service rate arrival rate and arrival rate /m. whereas, the average time of responsesare = 2 and m=10 for varied  showed in the diagram.

The instinctive description for systems behaviors are: - for 10 parallel cases of M/M/1 queues, there have always been, the probability of non-zero where some servers are idle and rest of them have plenty of customers. In opposite to that, it could not be happened in M/M/m case. The fat servers (single) are designed specifically for light loads than the M/M/10 systems. As when there are when k is greater than 10 clients in systems, where it has smaller rate of services k. The customers in the fat servers are served with services rate that are full, 10, =20.  




The law provides less significant relationship in between E (L), (mean customer numbers in system), and E (S) (Sojourn Time) as well as (average customer numbers entering in system/per unit).

Assume that all clients pay 1 dollar for every unit time while in the framework. It is expected that the framework's limit is adequate to manage the clients (i.e. the quantity of clients in the framework does not develop to boundlessness). Instinctively, this outcome can be comprehended as takes after. The primary plausibility is to let pay all clients "persistently". This cash can be earned in two ways. At that point the normal prize earned by the framework breaks even with E (L) dollar per unit time. In harmony, the normal number of clients leaving the framework 9 per unit time is equivalent to the normal number of clients entering the framework. The second probability is to let clients pay 1 dollar for every unit time for their home in the framework when they take off.  So the framework gains a normal prize of E(S) dollar per unit time.

For demonstrating the Little’s Law, we would consider significant queue model having one server. In this model, the relationships would be derived in terms of performance measures by the application of law in the defined systems. The application of the law consist server relations (yields). It also yields a relation while applying little law in the shape of waiting time W and queue length Lq, explicitly

Finally, when we apply Little's law to the server only, we obtain

< >PASTA PropertyThe Poisson arrival queuing systems for M/M/1 are very special properties holding average customers findings’ on eth same situation in queue system would be considered as an outside observance system looking at arbitrary times. More specifically, the customer fractions found in the system arrivals in A state are exactly similar fractions of time in A state. Such a property is only suitable for Poisson arrivals. Therefore, for making queue with the Poisson arrivals at arbitrary point in time average must be found.

In specific, there is no truth in property. For example, the D/D/1 system emptied at time 0 along with arrivals at 1, 3, 5, … as well as service time is 1. Every new customer that arrives would find an empty system, whereas the time fraction of system is half. Such property is called to be PASTA property, an acronym for Poisson Arrivals present in average times. On the other hand, the Poisson arrivals are occurred completely in random manners. The occurrences of arrivals are in random ways rather being standardized.


The point B, yellow circle represent on the graph as an outside source of customer. The graph contains three inter linked service centers as well as outside destinations, C where each have mean service time. Whenever a client leaves a center, he might have more than one possible destination placed as mark A in the graph. It is significant to describe the routing probability, which is done by the attachment of a weight to each destination possible. For instance, while leaving the second queue, there would be two possible destinations i.e. queue 0 and queue 1. If both of the destinations amounted to weight 1, then there would be an equal opportunity of leaving customer going for either queue. If queue 0 had, weight 2 and queue 1 weight 1, then on average three customers would fall in queue 1 and two would be falling in the queue 0.

There is no source or sink for the closed queue networks. The example could be seen in the below diagram. The performance of the service centers are in open network form and routing probability is explained in the similar manners (A on diagram). When there is closed network development, it is adequate to explain the each service center customers (Slater, 2000). Such customers could easily travel along the path of the network but could not leave it.          




Number in this aspect is about why that can accommodate with different perspective and show relevance with implementation. Telephone numbers are change able according to blocks and this change recorded with queuing system. The block division is known as best way to try to accomplish what is requiring as what is not so needy in any way. 10,000 blocks are being divided as per require assessment and show that how it is responsible for what kind of purpose.

The numbering assignments believe in queuing system and assure that how it can tackle issues related with employees and blockage purpose. Theoretically, it is possible that employees can manage ways with situations and situational analysis is possible with employees and factors effects it. There is some set of parameter and show each time evaluations is about to express with difference of 3 hours and that difference can increase with increasing value accordingly.

For example, in queuing system average is possible because if employee work in 9 hours a day and represents 5 applications on daily basis. Daily average is possible and represents as 3 hours for aggregate.


Event Helix. (2015, September 17). M/M/1 Queueing System. Retrieved from

Event Helix. (2015, September 16). Queueing Theory Basics. Retrieved from Event Helix:

Krondorf, M. (2006). An Introduction into Queueing Theory. TUD.

Rasing, A. (1990). Queueing Theory. EUT.

Slater, T. (2000, June). Queueing Networks. Retrieved from University of Edinburgh: :

Teorisi, K., SUCU, Y., & Simdekler, M. (2003). YÖNETİM BİLİŞİM SİSTEMLERİ BÖLÜMÜ . Trakya University.

Willig, A. (1999). A Short Introduction to Queueing Theory. TUB.

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