Questions and Answers of Essentials Of Stochastic Processes

(a) Find the stationary distribution for the transition probability 1 2 3 4 1 0 2=3 0 1=3 2 1=3 0 2=3 0 3 0 1=6 0 5=6 4 2=5 0 3=5 0 and show that it does not satisfy the detailed balance condition
Find the stationary distributions for the chains in exercises (a) 1.2, (b) 1.3, and (c) 1.7.
Find the stationary distributions for the Markov chains on f1; 2; 3; 4g with transition matrices:.a/0 BB@0:7 0 0:3 0 0:6 0 0:4 0 0 0:5 0 0:5 0 0:4 0 0:6 1CCA.b/0 BB@0:7 0:3 0 0 0:2 0:5 0:3 0 0 0:3
Find the stationary distributions for the Markov chains with transitionmatrices:.a/ 1 2 3 1 0:5 0:4 0:1 2 0:2 0:5 0:3 3 0:1 0:3 0:6.b/ 1 2 3 1 0:5 0:4 0:1 2 0:3 0:4 0:3 3 0:2 0:2 0:6.c/ 1 2 3 1 0:6
Consider the following transition matrices. Identify the transient and recurrent states, and the irreducible closed sets in the Markov chains. Give reasons for your answers..a/ 1 2 3 4 5 1 0:4 0:3
Suppose that the probability it rains today is 0.3 if neither of the last 2 days was rainy, but 0.6 if at least one of the last 2 days was rainy. Let the weather on day n;Wn, be R for rain, or S for
A taxicab driver moves between the airport A and two hotels B and C according to the following rules. If he is at the airport, he will be at one of the two hotels next with equal probability. If at a
Consider a gambler’s ruin chain with N D 4. That is, if 1 i 3; p.i; i C 1/ D 0:4, and p.i; i 1/ D 0:6, but the endpoints are absorbing states: p.0; 0/ D 1 and p.4; 4/ D 1 Compute p3.1; 4/ and
The 1990 census showed that 36% of the households in the District of Columbia were homeowners while the remainder were renters. During the next decade 6% of the homeowners became renters and 12% of
We repeated roll two four sided dice with numbers 1, 2, 3, and 4 on them. Let Yk be the sum on the kth roll, Sn D Y1 C C Yn be the total of the first n rolls, and Xn D Sn .mod 6/. Find the
Five white balls and five black balls are distributed in two urns in such a way that each urn contains five balls. At each step we draw one ball from each urn and exchange them. Let Xn be the number
A fair coin is tossed repeatedly with results Y0; Y1; Y2; : : : that are 0 or 1 with probability 1/2 each. For n 1 let Xn D Yn C Yn1 be the number of 1’s in the.n 1/th and nth tosses. Is Xn a
. Consider an M/G/1/N queue with multiple vacation and exhaustive service. Assume that the duration of vacation is of constant length V.Then the probabilitities n , that there are j customers in the
. Reliability analysis with breakdown and repair. Li et al. (1997) consider an M/G/1 queue with server breakdowns where service time X is general.They consider the situation that the server has an
. Kella (1989) considers an M/G/1 - V„, system with a threshold policy.Here the server goes on taking vacations of random length V until the first instance after a vacation where he finds at least
. M/G/1 system under N-policy and general start-up time.Suppose that instead of zero start-up time, the start-up times are IID random variables with common DF D(.) with mean uand LST D* (.) The
. M/G/1 queue under T-policy with zero start-up time.Here the facility is controlled in a different manner. Every time the server becomes idle after exhaustive service, the service facility is
. M/ G/1 under N-policy with zero start-up time.Suppose that demand for service arises in accordance with a Poisson process with rate A. The service times are IID random variable, with DF B having
. M/G(Q,)/1: average long-run cost. Assume that the input is Poisson with rate, and the service times are IID random variables, denoted by v, with moments bk = E(v). The service rule is general bulk
. Consider an M/G/1 system. Let W(t) be the virtual delay, and let a busy period start at t, with the arrival of a customer whose service time v has DF B() with LST B*() and moments bi. The delay
. Show that for a G/ M/1 queue E|zNe-il\ _ ME - uzØ1(-1)0 < z < 1, t real, us - u + it where ø1 is the characteristic function of the RV having DF A(u) =Pr[u, ≤ u) and § = {(z) is the unique
. Show that for an M/ M/1 queue 15 (2)E {z"e"] = " -il'where(+1) - v(2 +1)2 - 42uz]]Č(z) =2x Write down the PGF of N and the PDF of I (Prabhu, 1980).
. Distribution with bounded mean residual life. A nondiscrete distribution F is said to have its mean residual life bounded above (below) by y, denoted by y-MRLA (y-MRLB) iff[1- F(w)}du y for all +
. Marchal's weighting factors.As an approximation for E ( W), Marchal suggests some weighting factors to be used with the upper bound. The weighting factor p2 + 2202 1 + 1202(which tends to I as p
. Suppose that there is a random start-up time R after every idle period so that the first customer in every busy period suffers a random delay R before his service commences, and assume that the
. Batch service with fixed batch size k. Let Win be the waiting time in queue of the last (kth) person who arrived and formed the nth batch of k customers. Then show that the expected waiting time
. Find the expressions for the mean and the variance of the waiting time for the systems (i) E2/ M/1 and (ii) M/G/1.
. Show that W. = max(0, X, + X"-1+ ... + Xn-r+1,1 gr ≤ n).
. For the sample sequence { W", " ≥ 0} with first eight elements A ={ Wo = 0, W] > 0, W2 = 0, W3 > 0, W4 > 0, W5 = 0, W. > 0, W > 0}compute the Was in terms of S,s, and the Ins in terms of S,s, and
. Using Eq. (7.1.25), find the waiting-time distribution for the D/D/1 queue.
. Obtain the upper and lower bounds of E (W) for a D/ M/1 queue.
. For an M/ D/1 system, find K*(s), K(u); find E(X„) from K*(s).
. Show that the DF K(x) = P{X ,, ≤ x), X" = Un - u ,, for the system G/M/1 is given by 00 1K(x) =dA(u) {1 - e->(*+*), x x} = 1 - e-(y -* ), y > x > 0.(Prabhu, 1980)
. Using (7.5.14a) and (7.5.14b), show that the (two-sided) LST of the RV X„ = Un - u ,, for an M/ M/1 system is given by K*(s)(2 - 5) (M + s)Verify that K*(s) = B*(s) A*(-s).
. M/G/1 queue: interdeparture interval Let y be the interdeparture interval of an M/G/1 queue in steady state. Show that the LST of y is given by D'(s) =pB"(s) +(1-p){A*(s) B* (s)}, where A*(s),
. Busy Period of an M/G/oo Queue Define a busy period of an infinite-server queue with Poisson input as the interval during which at least one customer is present and is receiving service. Denote
. (a) Consider a G/G/1 system. Denote by R the arrival-epoch system size. Denote by N the general-time-system size. Denote for n =0,1,2,3 ,....an = Un = Pr(R = n)û, = Pr(R > n)Pn = Pr(N = n), Pn =
. Losses in an M/G/1/n queue with finite buffer.Righter (1999) discusses this problem, earlier considered by Abramov(1997).Let I ,, be the indicator variable that is 1 if there is a loss during a
. (a) Consider an M/G/1 finite queue with total space capacity n. Let To be the delay with DF H(t) and LST H*(s), let u be the service time with DF B(t) and LST B*(s), and let A ,, (s) be the LST of
. Show that (with notations as in Section 6.4.4)XE(12)p2 E(₸;) =(1 - p)3 E(TO) +(1 - p)2 E(T)XE(v2)E(TO)E(T.) = (1 - p)3 E (To) +(1 - p)2*(Miller, 1975)
. Consider an M/G/1 having an exceptional service for the first unit in a busy period (this may be necessitated for doing some extra work or for making some initial work to start services before
. Consider an M/ G/1 system with X ,, as defined in Eq. (6.3.1) an K(.) as given in Eq. (6.2.2). Let u(n) be defined by[inf{m ≥ 0: X"+m = 0}, if this set is nonempty;-p(m) =otherwise, n =0,1,2
. Higher moments of Wand Wa in an M/ G/1 queue: Recursive formulas for computation.Takagi and Sakamaki (1995) discuss symbolic moment calculation.Obtain the following recursive formulas:n b(+)(a)
. Show that the steady-state probabilities p;(=v;), j = 0, 1,2 ,... in an M/G/1 queue satisfy the recursive relation pj = Maj-1 po +À>aj-kPk j = 1,2 ,...,(A)i=1 where 8an=10 e-xr (ht)"-{1 - B(t)}dt,
. M/G/1 queue-length distribution (Willmott, 1988)(a) Refer to Section 6.3 and the Pollaczek-Khinchin formula (6.3.7).Verify that K(s) is a PGF; so also is K(s) -1 G(s) =_ 8"S" =(A)p(s-1)'where g ,,
. Suppose that the busy period T is initiated by m(≥1) customers(i.e ., there are m customers at the commencement of the busy period).Then the PGF of the number N„,(T) served during the busy
. Consider an M/G/1 system in steady state having service-time distri-bution B(.) with finite first- and second-order moments. Show that any one of the following statements implies the other two:(i)
. Consider an M/G/1 queueing system in steady state. Show that {k;} is geometric iff { p;}, j = 0, 1, 2, . .., is geometric. The result also holds for the zero-truncated geometric, that is, for j =
. Cycle-time distribution in cyclic exponential network.Consider an M-node cyclic network with K circulating jobs, the nodes having independent exponential servers with parameters ( ,, i = 1, 2 ,...,
. Sojourn-time distribution in cyclic exponential network.Consider a cyclic network with K circulating jobs among two nodes: 1 and 2. The service time at node i is exponential with parameter ( ,, i =
. M/M/I-PS: conditional delay Let F(x; t) = Pr {D ≤ x| S ={} be the DF of the conditional delay, given that total service requirement is of length t, and let F* (s; t) be its LST. Then, when p <
. M/G/1-PS model Consider an M/ G/1 queue where the queue discipline is processor shar-ing or time-sharing. This discipline implies that if there are already (n-1)customers in the system, then the
. M/ G/1/K queue as Cyclic Network. (Refer to the next chapter for an analysis of M/ G/I queues.)Consider a cyclic network with K circulating jobs and two nodes, with a single server at each of them.
. Consider a closed network with three nodes and K circulating jobs. Sup-pose that the service times at the nodes are independent exponential RVs with rates u1, U2, and u3, respectively. Suppose that
. Consider Jackson's open-network model with a single server at each of the k nodes. Let T and S denote, respectively, the total time spent in the system and the total service time received by a
. M/ M/1 Queue with Bernoulli feedback.Consider an M/ M/1 FCFS queue with Bernoulli feedback such that after completion of service, the job may leave the system with probability q or may be fed back
. Consider an open network such that arrivals to node 0 occur from outside in accordance with a Poisson process with rate À. After receiving service at node 0, the job (customer) may leave the
. Consider an open network with two nodes having a single exponential server at each of two nodes with service rates ui, i = 1,2. Suppose that arrivals to node 1 occur in accordance with a Poisson
. The system M/ M(a, b)/1.(a) Let Y be the number of customers that arrive during the period the server is idle with q (0 ≤ q ≤ a - 1) in the queue, and let Z be the number of customers present
. The model M/ M(1, b)/2.Define the busy period as the interval during which at least one of the servers remains busy with p1,0(0) = 1. Show that the LST b" (s) of the busy period is given by H[ (s +
. The model M/ M(a, b)/2.The busy period may be defined as the interval during which (i) both the servers remain busy or (ii) at least one server remains busy.Case (i): Show that the busy-period
. The model M/ M(1,b)/2 Show thatFind Poo(s). Suppose that (p = 1/2bu Show that the steady-state probabilities are as given in (4.5.15a) to (4.5.17). Pio(s) = [(s + 1 ) ], Po,o(s) ((s P2,0(s) (s + )
. Batch size in systems with general service rule For an M/M(a, b)/1 system, let w = 1/(A + u), p = 2/u. The distri-bution of service batch size Y is given by% = Pr(Y = a)p Po.o=1-[-(r - w) +
. Model M/ M(1, b; uk)/1, with mean service time 1/ut; for a batch of k.Distribution of queue size and number in the batch being served in Poisson queue with usual bulk service has been considered by
. For a continuation of Problem 4.8, consider waiting-time distribution in the delay system. Show that the distribution of waiting time W in the queue d+c-1 P(W> )=HE E Pi F.i=max(0,d+c-r)where r(d,
. M / M/c (Kabak, 1968) queue with rates À and / = 1 Consider (i) the delay system M' / M/c/00, where customers are al-lowed to wait, and (ii) the loss system M / M/c/c, where customers are lost
. M / M/00 queue. Show that given N(0) = i, the correlation coeffi-cient p between N(t) and N(0) equals e-"' and is independent of the arrival rate & and of the batch size X. Ast -> oo, p -> 0, as is
. Multiple Poisson bulk arrival (MPBA) system (Jensen et al. 1977). Con-sider that groups of size j arrive according to a Poisson process with intensity À }, { N; (t), t ≥ 0}, j = 1,2 ,..., m, and
. For a continuation of Problem 4.4, show that the generating functionDeduce that for Et = E1 = M-that is, for MX / M/1 M(o2 + a2 +a)E{N} = 2(u - xa)and for MX / D/1, i(o2 + a2 + à)(2a)2 E{N} = 2(u
. MX / Ek/1 system (Restrepo, 1965).Let the state of the system be denoted by (n, s), n = 0, 1, 2 ,..., s =1,2 ,..., k where n is the number of customers in the system and s is the number of phases
. The System MX / M/1(a) Find the variance of the number in the system for the MX / M/1 system.(b) Find the mean and variance of the number in the system for the MX / M/1 system where X has a
. The system Ex/ M/1 Write down the Chapman-Kolmogorov equations and obtain an expres-sion for P*(s,«).Obtain the relevant equations for finding the distribution of the busy period T (the interval
. The system M/ Ek/ 1: transient behavior. Define 00 00 P(s, t) => Pm(1)s", P*(s,a) = > pt(a)s",₪=0 where p*(@) is the LT of p,(t). Assume that po(0) = 1. Show that s - ku(1-s) po(@)P* (s,a) =s(c+
. M/ M/c queue with impatient customers Suppose that in an M/ M/c queue, each arriving customer enters the system but is only willing to wait in queue for a fixed time T > 0, after which the customer
. Transient behavior of M/ M/c/c Loss Model, This topic has received attention of late. Abate and Whitt (1998) discuss it by using numerical transform inversion.The rate of convergence of the M/
. Loss formula for M/ M/c/c + r,r ≥ 0 Model.This has been discussed by Pacheco (1994b). It can be shown that the loss formula for integral c and r becomes(a‘/c!)(a/c)"B(c, a; r) =Di=0 at/ k! +
. Multiserver Poisson queue with ordered entry (Nawijn, 1983). Consider the following two c-channel systems with ordered entry such that the c-channels are numbered 1, 2, ..., c and an arriving
. Consider a three-channel Poisson queue with ordered entry having no waiting space (as in Section 3.11.3). Find the steady-state probabilities and verify the results with those of the corresponding
. Multichannel queue with ordered entry and heterogeneous servers(Mastsui and Fukuta, 1977).Consider an ordered entry Poisson input queue (with rate A) with c exponential servers, the ith server
. M/ M/c queue (c ≥ 1): transient state distribution.Suppose that k(≥1) customers are already present at time to = 0 and that the nth new customer arrives at time t, (n ≥ 1).Denote X, = number
. Two-channel model with ordered entry and with M = 1, N = 3.Let P ;,; denote the probability that the number in channel 1 is i and that the number in channel 2 is j (i, j include those in service,
. Two-channel model with ordered entry with M = N = 1.Suppose that the service rates at the two channels are different, being ui and u2 at the first and second channel, respectively. Show that the
. Multiserver queue with balking and reneging. Consider a c-server queueing system with Poisson input with rate > and exponential service time with rate u for each of all c-servers. Suppose that (i)
. Transient output distribution for an M/ M/c system.Let R,(k | n, m) etc ., be defined as in Section 3.10.3, and let 00 R*(k| n, m) = R2(k| n, m) =1 e-z R, (k | n, m)dt Jo be the LT of R,(k | n, m).
. The transient state distribution of an M/ M/1 queue has been obtained in a different form (by Sharma) as given below.Show that the probability p(n, t) that there are n (customers) in the system (M/
. M/M/1: Two-dimensional state model Let the state of the system by time t be given by the ordered pair (i, j), where i is the number of arrivals and j is the number of departures by time t, and let
. Transient solution of an M/M/1 queue; alternative approach(Parthasarathy, 1987). Define qx(1) = {exp() + u) +}[upk(+) - Pk-1(+)], k= 1,2,=0, k=0 ,- 1 ,- 2 ...a = 2/2u.B == Vp 1-In(t) is a modified
. Consider a machine repair problem with c repairpeople, m machines(c < m), and exponential working time and repair time having ratesÀ and u, respectively. Suppose that m is very large. Show that
. Suppose that a machine breaks down, independently of others, in ac-cordance with a Poisson process, the average length of time for which a machine remains in working order being 36 hours. The
. M/ M/1 queue: waiting time in the system for an arrival at instant t,(virtual waiting time in the system). Show that = F(x, t) P{W, x|t} = 1-eux and the PDF of W, is given by (x) s! Pn(t), n=0s=0
. M/ M/c queue with servers' vacations (Levy and Yechiali, 1976).Consider an M/ M/c system in which a server proceeds on vacation when he has no unit to serve (the length of time he is on vacation
. Show that for the M/ M/c system in steady state, the PGF of the number in the queue is given by 1 - B"[A(1-z)/c]G(z) =B+[](1 - 2)/c] - 2Pc-1+ pi, i-0 where B*(s) = u/(u + s) is the LST of
. Show that the expected busy period for an M/ M/1/ K queueing system equals 1 - ak +1 E(T) =for a = -¥1.u(1-a)Show further that the expected number of loss during a busy period is 1 according as a
. Show that for the M/ M/1 queueing system starting with k customers at time 0, the joint distribution of the busy period Ti, and the number N(TR) served during the busy period initiated by k
. De Vany's Model (1976).De Vany uses the M/ M/1/K model to determine the effective demand function under the conditions: (1) the arrival stream is Poisson with rateÀ(p) (where p is price), (2)
. Naor's model for regulation of queue size (Naor, 1969).Suppose that the cost to a customer of staying in a queue (i.e ., for queue-ing) is c per unit time and that the reward collected at the end
. An M/M/1 queue with control-limit policy and exponential start-up time (Baker, 1973).Here the control policy is to turn off the system and withdraw the server when the system becomes empty and to
. Dufkova and Zitek's (1975) model of an M/ M/1 system with a class of queueing disciplines.Suppose that when the server becomes free, he accepts either the first in the queue with probability 8 or
. (a) Combination of service channels. Consider that two identical M/ M/1 queueing systems with the same rates À, u (intensity p = >/u) are in operation side by side (with separate queues) in a
. Renewal-reward process. Consider a renewal process { X1, n = 1,2 ,... }.Suppose that renewal epochs are to = 0, 1 ,..., and that N(t) is the num-ber of renewals by time t; associate a RV Y;(i = 1,2

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