Queuing Models

* QUEUING MODELS * •

Definition :A Queuing models can be described as composed of customers arriving for service waiting for service if it is not immediate and if having waited for service leaving the system after being served.

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Illustrations :   

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Care waiting at petrol pump for service. Customers waiting at bank. Telephone subscribers waiting for connection. Goods in production shop waiting for the Cranes.

Constituents of Queuing System :(1) Arrival pattern. (2) Service facility. (3) Queue discipline.  Arrival pattern :o It is average rate at which customers arrive as well as statistical pattern of arrival. o Regular pattern of arrival is not much easy to deal mathematically. o If no of potential customers is infinitely large then probability of an arrival in next interval of time will not depend upon no of customer already in the system. o When arrivals are completely random they follow poission distribution with mean equal to average no of arrivals per unit time. o There are several other types of the arrival patterns which we shall not deal due to restricted application.  Service Facility :(1) Availability of Service :o It is necessary to examine if there any constraints which reduce the no of customers that can be served at a time. o E.g. in waiting line of suburban train probability distribution of no. of passengers that can be accommodated in a train that arrives can be accommodated in a train that arrives in relevant apart from the timings of the train services. (2) No. of service centres :o If there is only one service centre referred to as a service channel. We can serve one at a time. o There are obviously more than one service centre and behariour of the queues will vary with no of channels available for service. o Multiple service channels may be arranged in service or in parallel. o Conters in bank are the examples of arrangement in series.

o Ticket booths in railway station have multiple channels with parallel arrangement.  Duration of Service :Time taken to serve a customer is called duration of service. o Constant service time. o Completely random service time. o Service time following Erlang distribution. •

Analysis of Queuing process :A typical investigation of queuing system comprise of steps as follows : Preliminary study.  Exploration of various alternatives.  Collection of date and analysis  Evaluation of alternatives.

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Objectives of Queuing Models : Reminisce queues in realm.  Analyze the queuing process mathematically.

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Queuing Discipline : (FCFS) :- First Came First Serve.  (LIFO) :- Last in First out. FCFS is used as a common pattern to be selected when the customers arrive. LIFO is used by the storekeeper when he has to carry on the removal and handling of the stocks.

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Customer Bahaviour :(1) Balking :- If arrival customers do not join a queue because of thieir reluctance to wait. (2) Collusion :- It happens when a single person joins the queue but demand service on behalf of several customers. (3) Reneging :- Customer do not wait beyond a certain time, get impatient and leave the queue. (4) Jockeying :- Some customers move from one queue to another in a multiple service centre. This is called jockeying.

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Multiple Service Channels – infinte pop ln -

P Po for n  c n n! P P for n  c = C!(cn c ) P 

1 -

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Po =

c 1 pn

  n 0 n!

pc

p  c! 1   c  c      E(m) =   po 2 C  1 ! C       c       E(n) =   po  2   C  1 ! C    c     1 E(v) =   po  2   C  1 ! C    c     E(w) =   po 2 C  1 ! C      

Notations :Queuing systems are denoted by M/M/1 or M/M/C M – exponential inter arrival distribution. M – Exponential service time distribution. C – No of Channels.  - Average no. arrival per unit time  - no of customers served by unit time. p – Traffic intensity. C – No of Service channels. M – no of customers in system. N – no of customers in system. Pn = steady state probability of finding n people in system. E(m) – Average length of queue. E(m/m>0) = Average length of non- empty queue. E (n) = Average no of customer in the system. W = Steady state waiting time of customers P (W = 0) = Probability of not having to wait in the queue. E (W/w>0) = Average waiting time of customers given that there is a queue V – waiting time in queue + Service time F(v) = Prob density function of time spont by a customer in system. E(V) = Average time spent by the customer in the system

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Single Channel queuing models :  Infinite population :-

Pn = (1-P) P n

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P0 = (1-P)

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E(m) =

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E(m/ m>0) =

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E(n) =

      

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   E(w) = (  ) 1 E(w/ w>0) =  P(w = 0) = 1-P 1 E(v) =  (1  P)c  w(  ) F(w) =

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F(v) =

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2

(   )c  v(  )

No of customers limited to N P 

1 P 1  P n 1  1 P  n P n 1  1  P 

Pn  

n = 0,1,2,…………….N P(w = O) = Po =

1 P 1  P n 1

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Erlang Family of distribution of service times. When in any queuing process if we get that service time follows negative exponential distribution we take S.D and mean both equal. When S.D. and mean differ then models have to be made more general by using a distribution which confirms closely practical problems but yet retains the simplicity of the properties of negative exponential distribution. As it was first studied by A.K. Erlang hence it was called “Erlang family of distribution” There are k phases. Average time taken by a customer through each phase is units. Then service

()  j1e t f (t)  (   1)! 1  1 S.D. =    1 Mode =  Negative exponential distribution. Mean =

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  1 2 Average queue length = E(m) = 2 (   )   1 2   Average no of units in system E(n) 2   (   )   1  Average waiting time E(w) = 2 (   )   1 2   Average item spent in system E(v) = 2  (   )     For 2 E(m) = 2(   ) 2   E(n) = 2(   )  2 E(w) = 2(  ) 2 1  E(v) = 2(  ) 

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Queuing Disciplines in finite queuing process (1) First Come – First Served (2) Priority (3) Random.

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Notations of finite queuing tables NPopulation MService Channels TAverage Service time W - Average Waiting time UAverage running time or mean time HAverage no of units being served. LAverage no of units waiting for service. JAverage no of units in operation. FEfficiency factor XService factor DProbability that if a unit calls for service, it will have to wait.

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Measures of system efficiency. N= X= H= L= L= F=

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J+H+L T T+U NT  FNX T+W+U NU  N(1  F) T+W+U NU  NF(1  X) T+W+U H+J T+U  H+L+J T+U+W

Uses of finite queuing Tables :Find mean service time T and mean running service factor. Compute service factor. Select table corresponding to population N For given Population, locate service factor value. Read off from the table value of D and F for the no of service crew M. if necessary values may be interpolated between relevant values of X. Calculate L,W,H,J from the formulae given.